Polyhedra golf ball with lower drag coefficient

ABSTRACT

A golf ball having an outer surface with a pattern forming a polyhedron. The pattern can be flat faces forming sharp edges and sharp points therebetween. In one embodiment, the polyhedron is a Goldberg polyhedron.

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No. 62/616,861, filed Jan. 12, 2018, the entire contents of which are incorporated herein by reference.

BACKGROUND Technical Field

The present invention relates to golf balls having a polyhedra design that can yield lower drag coefficient than a dimpled sphere. The drag reduction is applicable to other sports equipment in general having a bluff body, such as the head of a golf club or a bike helmet.

Background of the Related Art

For the past 100 years the vast majority of commercial golf balls have used designs with dimples. For the purpose of this invention, a dimple generally refers to any curved or spherical depression in the face or outer surface of the ball. The traditional golf ball, as readily accepted by the consuming public, is spherical with a plurality of dimples, where the dimples are generally depressions on the outer surface of a sphere. The vast majority of commercial golf balls use dimples that have a substantially spherical shape. Some examples of such golf balls can be found in U.S. Pat. Nos. 6,290,615, 6,923,736, and U.S. Publ. No. 20110268833.

It is well established that such depressions can lower the drag coefficient of a ball compared to that of a smooth sphere at the same speed. The drag coefficient is a dimensionless parameter that is used to quantify the drag force of resistance of an object in a fluid. The drag force is always opposite to the direction in which the object travels. The drag coefficient for a golf ball is defined as C_(D)=2*Fd/(ρ*U²*A), where Fd is the drag force, p is the density of the fluid in which the object is moving, U is the speed of the object and A is the cross sectional area. For a sphere the cross-sectional area is πD²/4, where D is the diameter of the ball.

FIG. 1 shows the variation of the drag coefficient, C_(D), as a function of Reynolds number, Re, for smooth and dimpled spheres. The data were obtained by performing wind tunnel experiments of non-spinning spheres. The Reynolds number is a dimensionless parameter used in fluid mechanics and is defined as Re=U*D/v, where v is the kinematic viscosity in which the object moves. For a smooth sphere the drag coefficient (shown by the solid black line) remains constant (C_(D)˜0.5) until the Reynolds number approaches a critical value (Re_(cr)˜300,000). At this point, which is usually referred to as drag crisis, C_(D) decreases rapidly and hits a minimum, which is an order of magnitude lower C_(D)˜0.08. With further increase in the Reynolds number the flow enters the post-critical regime characterized by turbulent boundary layers on the surface of the sphere. In this regime the drag coefficient rises slowly with increasing Reynolds number.

In dimpled spheres (shown by dashed and dotted black lines) the drag crisis happens at a much lower critical Reynolds number (Re<100,000). In general, a dimpled sphere will have a 50% or more reduction in drag compared to a smooth sphere in the range of Reynolds number from 100,000 to 250,000. A golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 60,000 at the end of the flight. It is therefore very important that a golf ball design can achieve a very low drag coefficient in that range. The critical value of the Reynolds number, as well as the attained minimum drag coefficient in the post-critical regime depend on the dimple geometry and arrangement. In general, as the aggregate dimple volume, measured as the sum of a dimple volume of each dimple of the plurality of dimples, decreases the critical Reynolds is increased and the drag coefficient in the post-critical regime increases. As the aggregate dimple volume approaches zero the drag curve approaches that of a smooth sphere. Drag is one of the two forces (the other being lift) that influence the aerodynamic performance of a golf ball. To increase the carry distance on a driver shot, the distance a golf ball travels during the flight, the dimple design is a balance between achieving the lowest critical Reynolds number and the lowest drag coefficient in the post-critical regime.

However, dimples are the only configuration known to provide a golf ball having a reduced drag coefficient.

SUMMARY

Accordingly, it is an object of the invention to provide a golf ball having minimal drag coefficient. It is a further object of the invention to provide a golf ball having reduced drag coefficient compared to a golf ball having only dimples. It is another object to provide a golf ball with minimal drag coefficient that is non-dimpled. In one embodiment, the golf ball has a plurality of flat faces with sharp edges and points that collectively form a polyhedron. These and other objects of the invention, as well as many of the intended advantages thereof, will become more readily apparent when reference is made to the following description, taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 shows a plot of the drag coefficient C_(D) vs Reynolds number Re for smooth and dimpled spheres. The solid black line represents a smooth sphere (Achenbach, 1972); the double-dashed lines represent a dimpled sphere (J. Choi, 2006); and the dash-dot lines represent a dimpled sphere (Harvey, 1976). The shaded area represents the typical range of Reynolds experience by a golf ball in flight during a driver shot (50,000-200,000).

FIG. 2(a) shows a golf ball in accordance with one embodiment of the invention.

FIG. 2(b) shows an outline of a golf ball.

FIG. 2(c) shows an outline of a golf ball with non-sharp rounded edges.

FIG. 2(d) shows an icosahedron, a well-known Platonic solid used to derive the golf ball 100.

FIG. 2(e) shows an example of splitting a hexagonal face into 6 triangular faces.

FIG. 3 shows the Goldberg polyhedron with 192 faces.

FIG. 4 is a graph of the drag coefficient versus Reynolds for the Goldberg polyhedra with 162 faces and 192 faces.

FIG. 5 is a graph showing the C_(D) versus Reynolds for a Goldberg polyhedron with 192 faces and a dimpled sphere.

FIG. 6 is a graph showing the C_(D) versus Reynolds for a Goldberg polyhedron with 162 faces and a dimpled sphere.

FIG. 7 shows a geodesic polyhedron made from 320 triangles.

FIG. 8 shows a geodesic cube with 174 faces.

FIG. 9 shows a polyhedron with 162 faces and 162 dimples.

FIG. 10 is a graph showing the drag coefficient C_(D) versus Reynolds number Re for a Goldberg polyhedron with 162 faces and a Goldberg polyhedron with 162 faces and 162 dimples.

FIG. 11 is a comparison of drag coefficient C_(D) versus Reynolds number for one of the embodiments of the present invention shown in FIG. 9 against a commercial ball Callaway Superhot.

FIG. 12 is an alternative embodiment of a golf ball based on an icosahedron with 312 faces and 312 spherical dimples.

FIG. 13 is a comparison of drag coefficient C_(D) versus Reynolds number Re for one of the embodiments of the present invention based on a polyhedron with 312 faces and 312 spherical dimples of FIG. 12 against a commercial golf ball Bridgestone Tour.

DETAILED DESCRIPTION

In describing certain illustrative, non-limiting embodiments of the invention illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected, and it is to be understood that each specific term includes all technical equivalents that operate in similar manner to accomplish a similar purpose. Several embodiments of the invention are described for illustrative purposes, it being understood that the invention may be embodied in other forms not specifically shown in the drawings.

The present invention is directed to a golf ball design based on polyhedra that can have reduced drag coefficient compared to a dimpled sphere. In one embodiment, a family of golf ball designs are made up of convex polyhedra whose vertices lie on a sphere. A polyhedron is a solid in three dimensions with flat polygonal faces, straight sharp edges and sharp corners or vertices.

FIG. 2(a) shows a golf ball 100 in accordance with one embodiment of the present invention. The golf ball 100 has a body 110 with an inner core and an outer shell with an outer surface 112. A plurality of faces 120 are formed in the outer surface, creating a pattern 116. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 124. Here, the golf ball 100 is a polyhedron with 162 polygons.

The body 110 defines a circumscribed sphere 102, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 104 of the body 110. The sphere and the diameter provide a reference for the size of the golf ball. The Rules of Golf, jointly governed by the R&A and the USGA, state that the diameter of a “conforming” golf ball cannot be any smaller than 1.680 inches. For the purpose of a golf ball the diameter of the circumscribed sphere is at least 1.68 in. The vertices 122 a, 122 b of the polyhedron are the only points 104 on the polyhedron that lie on the sphere. Any point along the edges 124 a, 124 b or on the faces 120 a, 120 b of the polygons lies below the surface of the circumscribed sphere.

The golf ball body 110 is a polyhedron that is made from first faces 120 a and second faces 120 b. As shown, the first faces 120 a have a first shape, namely pentagons, and the second faces 120 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 120 a and 150 hexagons 120 b (a hexagon-to-pentagon ratio of 12.5:1), each having corners or points 122 a, 122 b connected by boundaries such as straight lines or edges 124 a, 124 b. In various other embodiments, other quantities and/or ratios of such pentagons 120 a and hexagons 120 b can be used. However, the number of polygons and the angle between them determine when the drag coefficient will start to drop and how low it will become. In general, as the number of faces is increased the drag crisis occurs at higher Reynolds number and the drag coefficient decreases. The first and second faces 120 a, 120 b form the pattern 116.

The edges 124 are sharp, in that the faces are at an angle with respect to one another. FIG. 2(b) shows a cross sectional cut through the body 110 of ball 100 of FIG. 2(a) along the line 150. In this embodiment the edges 124 are sharp, in that the radius of curvature 140 of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use. FIG. 2(c) shows a cross section cut of two faces with rounded edges, whose radius of curvature is more than 0.001 D. The resulting edge is not sharp and the reduction in drag is not maximized, which can be detrimental to the aerodynamic performance of the golf ball as the shape would approach that of a smooth sphere. Both a sharp edge and a non-sharp edge is shown in that embodiment for illustrative purposes. The angle θ formed between two adjacent flat/planar faces 120 is always smaller than 180 degrees. The geometric shape of the embodiment illustrated in FIG. 2(a) falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 120 a and an adjacent hexagonal face 120 b is 166.215 degrees. The angle between two adjacent hexagon faces 120 b varies from 161.5 degrees to 162.0 degrees.

Each face 120 is immediately adjacent and touching a neighboring face 120, such that each edge 124 forms a border between two neighboring faces 120 and each point 122 is at the intersection of three neighboring faces 120. And, each point 122 is at an opposite end of each linear edge 124 and is at the intersection of three linear edges 124. Accordingly, there is no gap or space between adjacent neighboring faces 120, and the faces 120 are contiguous and form a single integral, continuous outer surface 112 of the ball 100.

A golf ball usually has a rubber core and at least one more layer surrounding the core. The pattern 116 is formed on the outermost layer. The pattern is based on an icosahedron shown in FIG. 2(d). The icosahedron 170 is a well-known convex polyhedron made up of 20 equilateral triangle faces 180, 12 vertices 182 and 30 edges 184. An icosahedron is one of the five regular Platonic solids, the other four being the cube, the tetrahedron, the octahedron and the dodecahedron (see https://en.wikipedia.org/wiki/Platonic solid).

In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 120 a of the golf ball 100 shown in FIG. 2(a) are centered on the vertices of an icosahedron. Therefore, a pair of 3 pentagons 120 a forms an equilateral triangular pattern 180. Along each of the edges of the triangles 180 there are 3 hexagons 120 b. Finally, inside each triangular pattern 180 there are three hexagons 120 b. The pentagons 120 a are all equilateral, that is the 5 edges 124 a all have the same length equal to 0.151 D, where D is the diameter of the circumscribed sphere. The hexagons 120 b are not equilateral and the lengths of the edges 124 b vary from 0.151 D to 0.1834 D.

Table 1 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 122 of golf ball 100. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 104 and are with respect to the center of the sphere. The golf ball 100 contains 320 vertices 122, 480 straight edges 120 and 162 polygonal faces 120.

The particular polyhedron can be any suitable configuration, and in one embodiment is a class of solids called Goldberg polyhedra that comprises convex polyhedra that are made entirely from a combination of rectangles, pentagons and hexagons. Goldberg polyhedra are derived from either an icosahedron (a convex polyhedron made from 12 pentagons), or an octahedron (a convex polyhedron made from 8 triangles) or a tetrahedron (a convex polyhedron made from 4 triangles). An infinite number of Goldberg polyhedra exist, as shown in https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra.

However, it will be recognized that the invention can utilize any convex polyhedron (that is, a polyhedron made up of polygons whose angle is less than 180 degrees), though in one embodiment on such polyhedron is a convex polyhedron with sharp edges, flat faces forming a single plane, and adjacent faces having an angle of less than 180 degrees between them.

The particular configuration (Goldberg polyhedral 162 faces and icosahedral symmetry) can only be achieved with a combination of hexagons and pentagons. However other geometries with around 162 faces may be possible to do using only pentagons or only hexagons. Other embodiments of the invention can include a pattern with various geometric configurations. For example, the pattern can be comprised of more or fewer of hexagons and pentagons than shown. Or it can comprise all hexagons, all pentagons, or no hexagons or pentagons but instead one or more other shapes or polyhedrals having flat faces and sharp edges. One other shape can be formed, for example, by splitting each hexagon into 6 triangles or each pentagon into 5 triangles, which provides a similar drag coefficient. One embodiment can include any of the Goldberg polyhedra with a combination of pentagons and hexagons or even a convex polyhedral made of triangles or squares.

It is further noted that flat faces give lower drag and have the uniqueness of not being dimples (curved indentations). The flat faces only provide points of the faces that lie on the circumscribed sphere. The sharp edges are defined by the angle between two adjacent faces. In addition, the edges forming the boundaries between the two adjacent faces are flat and not excessively rounded. An example of a non-sharp edge is shown in FIG. 2(c). The angle between the two edges is the same as is in FIG. 2(b) but it is rounded such that the edge is not sharp.

As the number of polygons increases (i.e. from 162 faces to 312 faces) the angle between the faces increases too and approaches 180 degrees. For the embodiments that are based purely on polyhedral shape such as those shown in FIGS. 2(a), 2(b) with 162 and 192 faces the range of angles is between 160 and 165 degrees. For the other embodiments in which dimples are added inside each face it is possible to go to as many as 312 faces and the angle between the faces can increase to 172 degrees. In one embodiment, the maximum angle could be close to 175 degrees and a range of angles between 160 and 175 degrees may be suitable for the purpose of a golf ball. Thus, a convex polyhedra is one having faces with angles substantially with the values and in the ranges noted herein.

In FIG. 2(a), the ratio of pentagons to hexagons is 12:150, though any suitable ratio can be provided. For example, out of the 150 hexagons one of the hexagons can be split into 6 triangles and a have a polyhedron with 12 pentagons, 149 hexagons and 6 triangles and obtain substantially the same drag coefficient. FIG. 2(e) illustrates how such a splitting can be performed on one of the hexagonal faces 120. A vertex 140 can be chose anywhere inside the hexagon 120. For illustrative purposes, the vertex 140 is near the center of the hexagon although any other location can be used. Six new edges 142 can be formed by connecting the each of the vertices 122 with the new vertex 140. A triangular face 144 is formed by one edge 124 of the hexagon and two adjacent edges 142. The exact shape of the faces making up the polyhedral can vary but one important feature of the polyhedral pattern is the angle between faces.

FIG. 3 shows an example of a golf ball in accordance with another embodiment of the invention. The golf ball 200 has a body 210 with an inner core and an outer shell with an outer surface 212. A plurality of faces 220 are formed in the outer surface, creating a pattern 216. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 224. Here, the golf ball 200 is a polyhedron with 192 polygons.

The body 210 defines a circumscribed sphere 202, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 204 of the body 210. The vertices 222 a, 222 b of the polyhedron are the only points 204 on the polyhedron that lie on the sphere. Any point along the edges 224 a, 224 b or on the face of the polygons lies below the surface of the circumscribed sphere.

The golf ball body 210 is a polyhedron that is made from first faces 220 a and second faces 220 b. As shown, the first faces 220 a have a first shape, namely pentagons, and the second faces 220 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 220 a and 180 hexagons 220 b (a hexagon-to-pentagon ratio of 15:1), each having corners or points 222 a, 222 b connected by boundaries such as straight lines or edges 224 a, 224 b. In various other embodiments, other quantities and/or ratios of such pentagons 220 a and hexagons 220 b can be used. The first and second faces 220 a, 220 b form the pattern 216. The edges 224 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.

The geometric shape of the embodiment illustrated in FIG. 3 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 220 a and an adjacent hexagonal face 220 b is 167.6 degrees. The angle between two adjacent hexagon faces 120 b varies from 163.4 degrees to 164.2 degrees. When comparing this embodiment with the golf ball 100 illustrated in FIG. 2 it is obvious that as the number of faces on a convex polyhedron increases the angle between faces increases too.

Each face 220 is immediately adjacent and touching a neighboring face 220, such that each edge 224 forms a border between two neighboring faces 220 and each point 222 is at the intersection of three neighboring faces 220. And, each point 222 is at an opposite end of each linear edge 224 and is at the intersection of three linear edges 224. Accordingly, there is no gap or space between adjacent neighboring faces 220, and the faces 220 are contiguous and form a single integral, continuous outer surface 212 of the ball 200.

The pattern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 220 a of the golf ball 200 shown in FIG. 3 are centered on the vertices of an icosahedron. Therefore, any pair of 3 pentagons 220 a form an equilateral triangle 280. The pentagons 220 a are all equilateral, that is the 5 edges 224 a all have the same length equal to 0.136 D, where D is the diameter of the circumscribed sphere. The hexagons 220 b are not equilateral and the lengths of the edges 224 b vary from 0.136 D to 0.168 D.

Table 2 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 220 of polyhedron of golf ball 200. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 204 and are with respect to the center of the sphere. The golf ball 200 contains 380 vertices 222, 570 straight edges 220 and 192 polygonal faces 220.

From a visual perspective the above designs of FIGS. 2, 3 have the unique characteristics of not having any dimples. From a utility perspective the behavior of the drag coefficient is very interesting. FIG. 4 shows a graph of the drag coefficient, C_(D), versus the Reynolds number, Re, for the polyhedron with 162 and 192 faces. The drag coefficient was obtained by wind tunnel experiments of non-spinning models. Overall the drag curve is qualitatively very similar to that of a dimpled sphere. Namely there is a drag crisis that occurs around Re=60,000. For the polyhedron with 162 faces C_(D) reaches a minimum value of 0.16 at Re=90,000 and remains almost constant as the Reynolds increases. For the polyhedron with 192 faces C_(D) reaches a minimum value of 0.14 at Re=110,000 and remains almost constant as the Reynolds increases.

The graph reveals that as the number of faces increases the drag crisis shifts towards a higher Reynolds number and the C_(D) in the post-critical regime decreases. This feature can be taken into advantage when designing a golf ball to tailor the needs of a golfer. For an amateur golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 180,000 at the beginning of the flight to 60,000 at the end of the flight. For a professional golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 80,000 at the end of the flight.

In other words, the range of Reynolds number experienced by a golf ball for an amateur golfer is lower than that for a professional golfer. A golf ball with more polygon faces might suit the needs of a professional golfer while a golf ball with less polygon faces might suit the needs of an amateur golfer.

The advantage of a design based on a convex polyhedron such as golf ball 200 with respect to a dimpled golf ball is now discussed. A comparison of the drag curve of the polyhedron with 192 faces, namely golf ball 200, against a dimpled sphere is shown in FIG. 5. The dimpled sphere has 322 spherical dimples and is representative of a commercial golf ball. The drag crisis for the polyhedron with 192 faces, namely golf ball 200, occurs at approximately the same range of Reynolds numbers as the dimpled sphere. The minimum C_(D) for both balls is reached at Re=110,000. For the dimpled sphere C_(D)=0.16 while for the golf ball 200 C_(D)=0.14, that is 12.5% drag reduction. At Re=140,000 C_(D)=0.174 for the dimpled sphere while for the golf ball 200 C_(D)=0.147, that is 15% drag reduction. Indeed, the drag coefficient for golf ball 200 illustrated in FIG. 3 is consistently lower than that of a dimpled golf ball in the range of Re=90,000-220,000. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.

A comparison of the drag curve of the golf ball 100 against the same dimpled sphere is shown in FIG. 6. While C_(D) in the post-critical regime is almost identical for the two balls, the drag crisis for the golf ball 200 occurs at a lower Reynolds number. As a result, C_(D) for Re<110,000 is consistently lower for the polyhedron with 192 faces than for the dimpled sphere. Thus, the embodiment of FIG. 2(a) has a lower drag coefficient than a dimpled sphere. A lower drag coefficient is important because it plays an important role in achieving more carry distance during a driver shot.

FIGS. 7, 8 are additional non-limiting embodiments of the invention. Those golf balls 300, 400 have similar structure as the embodiments of FIGS. 2, 3, and those structures have similar purpose. Those structures have been assigned a similar reference numeral and similar structure with the differences noted below. For example, FIG. 7 shows an example of a golf ball 300 with a body 310 with an inner core and an outer shell with an outer surface 312. A plurality of faces 320 are formed in the outer surface, creating a pattern 316. All the faces are formed in the outer surface 312 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 324 and corners vertices 322. The body 310 defines a circumscribed sphere 302, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 304 of the body 310. And FIG. 8 shows an example of a golf ball 400 with a body 410 with an inner core and an outer shell with an outer surface 412. A plurality of faces 420 are formed in the outer surface, creating a pattern 416. All the faces are formed in the outer surface 412 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 424 and corners or vertices 422. The body 410 defines a circumscribed sphere 402, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 404 of the body 410.

The embodiment shown in FIGS. 7, 8 are included for illustrative purposes as examples of convex polyhedral that are not made from pentagons or hexagons. Other convex polyhedra not made up of pentagons or hexagons can also be used for the design of a golf ball. In another embodiment of the present invention a convex polyhedron is shown in FIG. 7. The polyhedron belongs to a class of solids called Geodesic polyhedron which are derived from an icosahedron by subdividing each face into smaller faces using a triangular grid, and then applying a canonicalization algorithm to make the result more spherical. (see https://en.wikipedia.org/wiki/Geodesic_polyhedron). The vertices of the polyhedron are the only points on the polyhedron that lie on a sphere. Any point along the edges or on the face of the triangles lies below the surface of a circumscribed sphere. The polyhedron in FIG. 7 is made from 320 triangles but any geodesic polyhedron with an arbitrary number of triangles can be used as the design of a golf ball.

In another embodiment of the present invention a convex polyhedron is shown in FIG. 8. The polyhedron belongs to a class of solids called Geodesic cubes which are made from rectangular faces. A geodesic cube is a polyhedron derived from a cube by subdividing each face into smaller faces using a square grid, and then applying a canonicalization algorithm to make the result more spherical (see http://dmccooey.com/polyhedra/GeodesicCubes.html). The vertices of the polyhedron are the only points on the polyhedron that lie on a sphere. Any point along the edges or on the face of the rectangular faces lies below the surface of a circumscribed sphere. The polyhedron in FIG. 8 is made from 174 rectangular flat faces but any geodesic cube with an arbitrary number of faces can be used as the design of a golf ball.

It is important to note that the polyhedra described above and shown in FIGS. 2, 3, 7, 8 do not contain any dimples (i.e., curved or spherical depressions or indents), but instead have flat surfaces that lie in a plane. However, the polyhedra provide enhanced aerodynamic characteristics, drag coefficient being one of them, that can help increase the carry distance of the golf ball.

However, golf ball geometries that are based on any convex polyhedra and either include or omit dimples are contemplated. In particular, for any convex polyhedra at least one of the faces may include one or more dimples. For example, FIG. 9 shows an embodiment of a golf ball 500 that is based on a convex polyhedron with a plurality of polygonal faces 520. The convex polyhedron is identical to the one shown in FIG. 2, and includes a body 510 with an inner core and an outer shell with an outer surface 512. A plurality of faces 520 are formed in the outer surface, creating a pattern 516. All the faces are formed in the outer surface 512 and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 524 and corners or vertices 522. The body 510 defines a circumscribed sphere 502, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 504 of the body 510.

However, in this embodiment each face 520 of the polyhedron contains one dimple 560. The dimples 560 have a substantially spherical shape and are created by subtracting spheres 570 from each of the faces 510 of the polyhedron 500. Table 3 lists the coordinates x, y and z of the center of the spheres 570 along with the diameter d of the spheres 570. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 504. The coordinates x, y, and z are with respect to the center of the circumscribed sphere. The total dimple volume ratio, defined as the aggregate volume removed from the surface of the polyhedron divided by the volume of the polyhedron, is 0.317%.

As shown, the dimples 560 are each located at the center of each 510. However, the dimples 560 can be positioned at another location such as offset within each face 510, or overlapping two or more faces 510. In addition, while spherical dimples 560 are shown, the dimples 560 can have any suitable size and shape. For example, non-spherical depressions can be utilized, such as triangles, hexagons, pentagons. And the dimples need not all have the same size and shape, for example there can be dimples with more than one size and more than one shape.

The effect that the addition of dimples has on the drag coefficient is now discussed. A comparison of the drag curve of this embodiment against that of golf ball 100 is shown in FIG. 10. There are two important observations. First when dimples are added to the faces of a polyhedron, the drag crisis occurs at a lower Reynolds number range. That is, the drag coefficient starts to drop at a lower Reynolds number. Second, the drag coefficient in the post-critical regime increases. This effect may be desirable when designing a golf ball for players with lower swing speeds such as an amateur golf player where the range of Reynolds number that the golf ball experiences during a driver shot is reduced. As the total dimple volume approaches zero the drag curve of golf ball 500 would approach that of golf ball 100. Therefore, one can fine tune the exact behavior of the drag curve by adjusting the total dimple volume. Each dimple is formed by subtracting a sphere for the face of the polyhedron as explained above. The dimple volume is the amount of volume that each sphere subtracts from the volume of the polyhedron.

FIG. 11 compares the drag curve of the golf ball 500 shown in FIG. 9 against a commercial golf ball, namely the Callaway SuperHot, which is a dimpled ball marketed as having a low drag coefficient and shown in U.S. Pat. No. 6,290,615. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the golf ball 500. The drag crisis for the golf ball 500 happens earlier, that is the drag coefficient starts to drop at lower Reynolds number. Clearly C_(D) for golf ball 500 is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 60,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances. The golf ball 500 is the exact same polyhedron of golf ball 100, however the golf ball 500 has a dimple on each face. The polyhedron which golf ball 500 is based on is identical to the polyhedron of golf ball 100 in FIG. 2.

FIG. 12 shows an example of a golf ball based on a convex polyhedron with dimples in accordance with another embodiment of the invention. The golf ball 600 has a body 610 with an inner core and an outer shell with an outer surface 612. A plurality of faces 620 are formed in the outer surface, creating a pattern 616. All the faces are formed in the outer surface and each of such faces is flat and lies in a plane. The faces are bound by straight or linear edges 624. Here, the golf ball 600 is based on a polyhedron with 312 polygons.

The body 610 defines a circumscribed sphere 602, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 604 of the body 610. The vertices 622 a, 622 b of the polyhedron are the only points 604 on the polyhedron that lie on the sphere. Any point along the edges 624 or on the face of the polygons lies below the surface of the circumscribed sphere.

The golf ball body 610 is a polyhedron that is made from first faces 620 a and second faces 620 b. As shown, the first faces 620 a have a first shape, namely pentagons, and the second faces 620 b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 620 a and 300 hexagons 620 b (a hexagon-to-pentagon ratio of 25:1), each having corners or points 622 a, 622 b connected by boundaries such as straight lines or edges 624 a, 624 b. The first and second faces 620 a, 620 b form the pattern 616. The edges 624 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.

The geometric shape of the embodiment illustrated in FIG. 12 falls into the class of convex polyhedral where the angle between any pair of adjacent faces is always less than 180 degrees. In this embodiment, the angle between a pentagon face 620 a and an adjacent hexagonal face 620 b is 170.2 degrees. The angle between two adjacent hexagon faces 620 b varies from 167.1 degrees to 168.2 degrees.

Each face 620 is immediately adjacent and touching a neighboring face 620, such that each edge 624 forms a border between two neighboring faces 620 and each point 622 is at the intersection of three neighboring faces 620. And, each point 622 is at an opposite end of each linear edge 624 and is at the intersection of three linear edges 624. Accordingly, there is no gap or space between adjacent neighboring faces 620, and the faces 620 are contiguous and form a single integral, continuous outer surface 612 of the ball 600.

The pattern is based on an icosahedron shown in FIG. 2(c). In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 620 a of the golf ball 600 shown in FIG. 12 are centered on the vertices of an icosahedron. Therefore, any pair of 3 pentagons 620 a form an equilateral triangle 680 shown with dashed line in FIG. 12. The pentagons 620 a are all equilateral, that is the 5 edges 624 a all have the same length equal to 0.102 D, where D is the diameter of the circumscribed sphere. The hexagons 620 b are not equilateral and the lengths of the edges 624 b vary from 0.102 D to 0.132 D.

Table 4 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the polyhedron. Table 4 lists the coordinates x, y, and z of all of the vertices 622 of polyhedron of golf ball 600. Faces are constructed by connecting the group of vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 604 and are with respect to the center of the sphere. The golf ball 600 contains 620 vertices 622, 930 straight edges 624 and 312 polygonal faces 620.

Each of the face 620 of the polyhedron contains one dimple 690. The dimples 690 have a substantially spherical shape and are created by subtracting spheres 670 from the face 620 of the polyhedron. Table 5 lists the coordinates x, y and z of the center of the spheres 670 along with the diameter d of the spheres 670. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 604. The coordinates x, y, and z are with respect to the center of the circumscribed sphere.

A comparison of the drag curve of this embodiment against that of a commercial golf ball, namely the Bridgestone Tour is shown in FIG. 13, which is a dimpled ball marketed as having a low drag so that the ball travels further, and described in U.S. Pat. No. 7,503,857. The graphs shows the invention having a lower drag coefficient. The drag coefficient for both balls were obtained by performing wind tunnel experiments using the actual commercial golf ball and a 3D printed prototype of the embodiment. The drag crisis for both golf balls occurs at approximately the same range of Reynolds number, namely from Re=50,000-80,000. At Reynolds number of 100,000 C_(D) for the Bridgestone Tour ball is 0.195 while C_(D) for the current embodiment is 0.162, a drag reduction of 17%. Overall C_(D) for the current embodiment is consistently lower than that of the commercial golf ball for the range of Reynolds numbers 80,000-160,000. A lower drag coefficient can help a golf ball achieve longer carry distances.

Other sizes and shapes of dimples are contemplated by the present invention, and any suitable size and shape dimple can be utilized in the golf balls of FIGS. 2-3, 7-9, 12. While FIG. 2 employs a convex polyhedron that belongs to a class of Goldberg polyhedral, FIGS. 9, 12 are revisions to the Goldberg principle in that each face of the polyhedron contains a dimple.

The following documents are incorporated herein by reference. Achenbach, E. (1972). Experiments on the flow past spheres at high Reynolds numbers. Journal of Fluid Mechanics. Harvey, P. W. (1976). Golf ball aerodynamics. Aeronautical Quarterly. J. Choi, W. J. (2006). Mechanism of drag reduction by dimples on a sphere. Physics of Fluids, 149-167. Ogg, S. S. (2001).

It is further noted that the description and claims use several positional, geometric or relational terms, such as neighboring, circular, spherical, round, and flat. Those terms are merely for convenience to facilitate the description based on the embodiments shown in the figures. Those terms are not intended to limit the invention. Thus, it should be recognized that the invention can be described in other ways without those geometric, relational, directional or positioning terms. In addition, the geometric or relational terms may not be exact. For instance, edges may not be exactly linear, hexagonal, pentagonal or spherical, but still be considered to be substantially linear, hexagonal, pentagonal or spherical, and faces may not be exactly flat or planar but still be considered to be substantially flat or planar because of, for example, roughness of surfaces, tolerances allowed in manufacturing, etc. And, other suitable geometries and relationships can be provided without departing from the spirit and scope of the invention.

In addition, while the invention has been shown and described with boundaries formed by straight edges 124, 224, 324, 424, 524, 624 and points 122, 222, 322, 422, 522, other suitable boundaries can be provided such as curves with or without rounded points. And the boundaries need not be sharped, but can be curved. And other suitable shapes can be utilized for the faces. And, while the invention has been described with a certain number of faces and/or dimples, other suitable numbers of faces and/or dimples, more or fewer, can also be provided within the spirit and scope of the invention. Within this specification, the various sizes, shapes and dimensions are approximate and exemplary to illustrate the scope of the invention and are not limiting. The faces need not all be the same shape and/or size, and there can be multiple sizes and shapes of faces.

The sizes and the terms “substantially” and “about” mean plus or minus 15-20%, and in one embodiment plus or minus 10%, and in other embodiments plus or minus 5%, and plus or minus 1-2%. In addition, while specific dimensions, sizes and shapes may be provided in certain embodiments of the invention, those are simply to illustrate the scope of the invention and are not limiting. Thus, other dimensions, sizes and/or shapes can be utilized without departing from the spirit and scope of the invention.

Use of the term optional or alternative with respect to any element of a claim means that the element is required, or alternatively, the element is not required, both alternatives being within the scope of the claim. Use of broader terms such as comprises, includes, and having may be understood to provide support for narrower terms such as consisting of, consisting essentially of, and comprised substantially of. Accordingly, the scope of protection is not limited by the description set out above but is defined by the claims that follow, that scope including all equivalents of the subject matter of the claims. Each and every claim is incorporated as further disclosure into the specification and the claims are embodiment(s) of the present disclosure.

TABLE 1 Vertex x/D y/D z/D Face Group of vertices 1 0.0000 0.0000 0.9778 1 301 300 69 296 295 20 2 0.6519 0.0000 0.7288 2 312 311 74 302 301 20 3 −0.3260 0.5646 0.7288 3 295 294 79 313 312 20 4 −0.3259 −0.5646 0.7288 4 299 300 301 302 303 304 5 0.7288 0.5646 0.3259 5 293 294 295 296 297 298 6 0.7288 −0.5645 0.3259 6 310 311 312 313 314 315 7 −0.8533 0.3489 0.3259 7 300 299 70 292 291 69 8 0.1245 0.9135 0.3259 8 303 302 74 308 307 73 9 0.1245 −0.9135 0.3259 9 297 296 69 291 290 68 10 −0.8533 −0.3489 0.3260 10 311 310 75 309 308 74 11 0.8533 0.3489 −0.3260 11 294 293 80 320 319 79 12 0.8533 −0.3489 −0.3259 12 314 313 79 319 318 78 13 −0.7288 0.5645 −0.3259 13 288 289 290 291 292 14 −0.1245 0.9135 −0.3259 14 305 306 307 308 309 15 −0.1245 −0.9135 −0.3259 15 316 317 318 319 320 16 −0.7288 −0.5646 −0.3259 16 244 243 70 299 304 17 17 0.3259 0.5646 −0.7288 17 304 303 73 251 250 17 18 0.3260 −0.5646 −0.7288 18 298 297 68 268 267 18 19 −0.6519 0.0000 −0.7288 19 261 260 80 293 298 18 20 0.0000 0.0000 −0.9778 20 278 277 75 310 315 19 21 0.1755 0.3040 0.9230 21 315 314 78 285 284 19 22 0.4845 0.3040 0.8050 22 288 292 70 243 242 66 23 0.5210 0.5716 0.6140 23 252 251 73 307 306 72 24 0.2345 0.7370 0.6140 24 269 268 68 290 289 67 25 0.0210 0.5716 0.8050 25 305 309 75 277 276 71 26 −0.3510 0.0000 0.9230 26 316 320 80 260 259 76 27 −0.5055 0.2676 0.8050 27 286 285 78 318 317 77 28 −0.7555 0.1654 0.6140 28 183 182 67 289 288 66 29 −0.7555 −0.1654 0.6140 29 206 205 72 306 305 71 30 −0.5055 −0.2676 0.8050 30 229 228 77 317 316 76 31 0.1755 −0.3040 0.9230 31 250 249 53 245 244 17 32 0.0210 −0.5716 0.8050 32 267 266 59 262 261 18 33 0.2345 −0.7370 0.6140 33 284 283 63 279 278 19 34 0.5210 −0.5716 0.6140 34 242 243 244 245 246 247 35 0.4845 −0.3040 0.8050 35 248 249 250 251 252 253 36 0.8770 0.0000 0.4540 36 265 266 267 268 269 270 37 0.9135 −0.2676 0.2630 37 276 277 278 279 280 281 38 0.9725 −0.1654 −0.0460 38 259 260 261 262 263 264 39 0.9725 0.1654 −0.0460 39 282 283 284 285 286 287 40 0.9135 0.2676 0.2630 40 184 183 66 242 247 11 41 −0.4385 0.7595 0.4540 41 253 252 72 205 204 14 42 −0.2250 0.9249 0.2630 42 270 269 67 182 181 12 43 −0.3430 0.9249 −0.0460 43 207 206 71 276 281 13 44 −0.6295 0.7595 −0.0460 44 230 229 76 259 264 15 45 −0.6885 0.6573 0.2630 45 287 286 77 228 227 16 46 −0.4385 −0.7595 0.4540 46 246 245 53 239 238 52 47 −0.6885 −0.6573 0.2630 47 249 248 54 240 239 53 48 −0.6295 −0.7595 −0.0460 48 266 265 60 258 257 59 49 −0.3430 −0.9249 −0.0460 49 263 262 59 257 256 58 50 −0.2250 −0.9249 0.2630 50 280 279 63 273 272 62 51 0.6295 0.7595 0.0460 51 283 282 64 274 273 63 52 0.6885 0.6573 −0.2630 52 179 180 181 182 183 184 53 0.4385 0.7595 −0.4540 53 202 203 204 205 206 207 54 0.2250 0.9249 −0.2630 54 225 226 227 228 229 230 55 0.3430 0.9249 0.0460 55 247 246 52 188 187 11 56 0.6295 −0.7595 0.0460 56 198 197 54 248 253 14 57 0.3430 −0.9249 0.0460 57 175 174 60 265 270 12 58 0.2250 −0.9249 −0.2630 58 281 280 62 211 210 13 59 0.4385 −0.7595 −0.4540 59 264 263 58 234 233 15 60 0.6885 −0.6573 −0.2630 60 221 220 64 282 287 16 61 −0.9725 0.1654 0.0460 61 237 238 239 240 241 62 −0.9135 0.2676 −0.2630 62 254 255 256 257 258 63 −0.8770 0.0000 −0.4540 63 271 272 273 274 275 64 −0.9135 −0.2676 −0.2630 64 187 186 39 179 184 11 65 −0.9725 −0.1654 0.0460 65 181 180 38 176 175 12 66 0.7555 0.1654 −0.6140 66 204 203 43 199 198 14 67 0.7555 −0.1654 −0.6140 67 210 209 44 202 207 13 68 0.5055 −0.2676 −0.8050 68 233 232 49 225 230 15 69 0.3510 0.0000 −0.9230 69 227 226 48 222 221 16 70 0.5055 0.2676 −0.8050 70 189 188 52 238 237 51 71 −0.5210 0.5716 −0.6140 71 197 196 55 241 240 54 72 −0.2345 0.7370 −0.6140 72 254 258 60 174 173 56 73 −0.0210 0.5716 −0.8050 73 235 234 58 256 255 57 74 −0.1755 0.3040 −0.9230 74 212 211 62 272 271 61 75 −0.4845 0.3040 −0.8050 75 220 219 65 275 274 64 76 −0.2345 −0.7370 −0.6140 76 180 179 39 171 170 38 77 −0.5210 −0.5716 −0.6140 77 203 202 44 194 193 43 78 −0.4845 −0.3040 −0.8050 78 226 225 49 217 216 48 79 −0.1755 −0.3040 −0.9230 79 185 186 187 188 189 190 80 −0.0210 −0.5716 −0.8050 80 196 197 198 199 200 201 81 0.2485 0.4305 0.8677 81 173 174 175 176 177 178 82 0.3932 0.4305 0.8124 82 208 209 210 211 212 213 83 0.4103 0.5558 0.7230 83 231 232 233 234 235 236 84 0.2762 0.6332 0.7230 84 219 220 221 222 223 224 85 0.1762 0.5558 0.8124 85 237 241 55 105 104 51 86 0.0023 0.3047 0.9342 86 160 159 57 255 254 56 87 −0.0798 0.4470 0.8714 87 271 275 65 134 133 61 88 −0.2538 0.4396 0.8361 88 186 185 40 172 171 39 89 −0.3472 0.2926 0.8714 89 177 176 38 170 169 37 90 −0.2650 0.1503 0.9342 90 200 199 43 193 192 42 91 −0.0895 0.1550 0.9616 91 209 208 45 195 194 44 92 0.2627 −0.1544 0.9342 92 232 231 50 218 217 49 93 0.4270 −0.1544 0.8714 93 223 222 48 216 215 47 94 0.5077 0.0000 0.8361 94 190 189 51 104 109 5 95 0.4270 0.1544 0.8714 95 106 105 55 196 201 8 96 0.2627 0.1544 0.9342 96 161 160 56 173 178 6 97 0.1790 0.0000 0.9616 97 236 235 57 159 158 9 98 0.8203 0.1503 0.5196 98 213 212 61 133 138 7 99 0.8397 0.2926 0.4181 99 135 134 65 219 224 10 100 0.7466 0.4396 0.4540 100 168 169 170 171 172 101 0.6405 0.4470 0.5963 101 191 192 193 194 195 102 0.6211 0.3047 0.6979 102 214 215 216 217 218 103 0.7078 0.1550 0.6571 103 100 99 40 185 190 5 104 0.5551 0.7856 0.2006 104 201 200 42 113 112 8 105 0.4028 0.8735 0.2006 105 178 177 37 165 164 6 106 0.2971 0.8664 0.3433 106 129 128 45 208 213 7 107 0.3477 0.7781 0.4890 107 152 151 50 231 236 9 108 0.5000 0.6902 0.4890 108 224 223 47 142 141 10 109 0.6018 0.6905 0.3433 109 104 105 106 107 108 109 110 −0.0467 0.6902 0.6979 110 156 157 158 159 160 161 111 0.0669 0.7781 0.5963 111 133 134 135 136 137 138 112 0.0075 0.8664 0.4540 112 168 172 40 99 98 36 113 −0.1664 0.8735 0.4181 113 166 165 37 169 168 36 114 −0.2800 0.7856 0.5196 114 114 113 42 192 191 41 115 −0.2196 0.6905 0.6571 115 191 195 45 128 127 41 116 −0.4971 0.0000 0.8677 116 214 218 50 151 150 46 117 −0.5694 0.1253 0.8124 117 143 142 47 215 214 46 118 −0.6865 0.0774 0.7230 118 109 108 23 101 100 5 119 −0.6865 −0.0774 0.7230 119 112 111 24 107 106 8 120 −0.5694 −0.1253 0.8124 120 164 163 34 156 161 6 121 −0.2650 −0.1503 0.9342 121 138 137 28 130 129 7 122 −0.3472 −0.2926 0.8714 122 158 157 33 153 152 9 123 −0.2538 −0.4396 0.8361 123 141 140 29 136 135 10 124 −0.0798 −0.4470 0.8714 124 98 99 100 101 102 103 125 0.0023 −0.3047 0.9342 125 110 111 112 113 114 115 126 −0.0895 −0.1550 0.9616 126 162 163 164 165 166 167 127 −0.5403 0.6353 0.5196 127 127 128 129 130 131 132 128 −0.6733 0.5809 0.4181 128 150 151 152 153 154 155 129 −0.7540 0.4267 0.4540 129 139 140 141 142 143 144 130 −0.7073 0.3312 0.5963 130 167 166 36 98 103 2 131 −0.5744 0.3855 0.6979 131 115 114 41 127 132 3 132 −0.4881 0.5354 0.6571 132 144 143 46 150 155 4 133 −0.9579 0.0879 0.2007 133 108 107 24 84 83 23 134 −0.9579 −0.0880 0.2007 134 157 156 34 148 147 33 135 −0.8988 −0.1759 0.3433 135 137 136 29 119 118 28 136 −0.8477 −0.0880 0.4890 136 102 101 23 83 82 22 137 −0.8477 0.0880 0.4890 137 111 110 25 85 84 24 138 −0.8988 0.1759 0.3433 138 163 162 35 149 148 34 139 −0.5744 −0.3855 0.6979 139 131 130 28 118 117 27 140 −0.7073 −0.3312 0.5963 140 154 153 33 147 146 32 141 −0.7540 −0.4267 0.4540 141 140 139 30 120 119 29 142 −0.6733 −0.5809 0.4181 142 103 102 22 95 94 2 143 −0.5403 −0.6353 0.5196 143 94 93 35 162 167 2 144 −0.4881 −0.5354 0.6571 144 88 87 25 110 115 3 145 0.2485 −0.4305 0.8677 145 132 131 27 89 88 3 146 0.1762 −0.5558 0.8124 146 155 154 32 124 123 4 147 0.2762 −0.6332 0.7230 147 123 122 30 139 144 4 148 0.4103 −0.5558 0.7230 148 81 82 83 84 85 149 0.3932 −0.4305 0.8124 149 145 146 147 148 149 150 −0.2800 −0.7856 0.5196 150 116 117 118 119 120 151 −0.1664 −0.8735 0.4181 151 96 95 22 82 81 21 152 0.0075 −0.8664 0.4540 152 81 85 25 87 86 21 153 0.0669 −0.7781 0.5963 153 145 149 35 93 92 31 154 −0.0467 −0.6902 0.6979 154 90 89 27 117 116 26 155 −0.2196 −0.6905 0.6571 155 125 124 32 146 145 31 156 0.5000 −0.6902 0.4890 156 116 120 30 122 121 26 157 0.3477 −0.7781 0.4890 157 92 93 94 95 96 97 158 0.2971 −0.8664 0.3433 158 86 87 88 89 90 91 159 0.4028 −0.8735 0.2007 159 121 122 123 124 125 126 160 0.5551 −0.7856 0.2007 160 97 96 21 86 91 1 161 0.6018 −0.6905 0.3433 161 126 125 31 92 97 1 162 0.6211 −0.3047 0.6979 162 91 90 26 121 126 1 163 0.6405 −0.4470 0.5963 164 0.7466 −0.4396 0.4540 165 0.8397 −0.2926 0.4181 166 0.8203 −0.1503 0.5196 167 0.7078 −0.1550 0.6571 168 0.9490 0.0000 0.3154 169 0.9660 −0.1253 0.2259 170 0.9937 −0.0774 0.0813 171 0.9937 0.0774 0.0813 172 0.9660 0.1253 0.2259 173 0.7492 −0.6353 0.0271 174 0.7806 −0.5809 −0.1372 175 0.8647 −0.4267 −0.1643 176 0.9247 −0.3312 −0.0271 177 0.8934 −0.3855 0.1372 178 0.8019 −0.5354 0.1643 179 0.9579 0.0880 −0.2007 180 0.9579 −0.0879 −0.2007 181 0.8988 −0.1759 −0.3433 182 0.8477 −0.0880 −0.4890 183 0.8477 0.0880 −0.4890 184 0.8988 0.1759 −0.3433 185 0.8934 0.3855 0.1372 186 0.9247 0.3312 −0.0271 187 0.8647 0.4267 −0.1643 188 0.7805 0.5809 −0.1372 189 0.7492 0.6353 0.0271 190 0.8019 0.5354 0.1643 191 −0.4745 0.8218 0.3154 192 −0.3745 0.8993 0.2259 193 −0.4298 0.8993 0.0813 194 −0.5639 0.8218 0.0813 195 −0.5915 0.7740 0.2259 196 0.1756 0.9664 0.0271 197 0.1128 0.9664 −0.1372 198 −0.0628 0.9622 −0.1643 199 −0.1756 0.9664 −0.0271 200 −0.1128 0.9664 0.1372 201 0.0628 0.9622 0.1643 202 −0.5551 0.7856 −0.2007 203 −0.4028 0.8735 −0.2007 204 −0.2971 0.8664 −0.3433 205 −0.3477 0.7781 −0.4890 206 −0.5000 0.6902 −0.4890 207 −0.6018 0.6905 −0.3433 208 −0.7806 0.5809 0.1372 209 −0.7492 0.6353 −0.0271 210 −0.8019 0.5354 −0.1643 211 −0.8934 0.3855 −0.1372 212 −0.9247 0.3312 0.0271 213 −0.8647 0.4267 0.1643 214 −0.4745 −0.8218 0.3154 215 −0.5915 −0.7740 0.2259 216 −0.5639 −0.8218 0.0813 217 −0.4298 −0.8993 0.0813 218 −0.3745 −0.8993 0.2259 219 −0.9247 −0.3312 0.0271 220 −0.8934 −0.3855 −0.1372 221 −0.8019 −0.5354 −0.1643 222 −0.7492 −0.6353 −0.0271 223 −0.7805 −0.5809 0.1372 224 −0.8647 −0.4267 0.1643 225 −0.4028 −0.8735 −0.2006 226 −0.5551 −0.7856 −0.2006 227 −0.6018 −0.6905 −0.3433 228 −0.5000 −0.6902 −0.4890 229 −0.3477 −0.7781 −0.4890 230 −0.2971 −0.8664 −0.3433 231 −0.1128 −0.9664 0.1372 232 −0.1756 −0.9664 −0.0271 233 −0.0628 −0.9622 −0.1643 234 0.1128 −0.9664 −0.1372 235 0.1756 −0.9664 0.0271 236 0.0628 −0.9622 0.1643 237 0.5639 0.8218 −0.0813 238 0.5915 0.7740 −0.2259 239 0.4745 0.8218 −0.3154 240 0.3745 0.8993 −0.2259 241 0.4298 0.8993 −0.0813 242 0.7073 0.3312 −0.5963 243 0.5744 0.3855 −0.6979 244 0.4881 0.5354 −0.6571 245 0.5403 0.6353 −0.5196 246 0.6733 0.5809 −0.4181 247 0.7540 0.4267 −0.4540 248 0.1664 0.8735 −0.4181 249 0.2800 0.7856 −0.5196 250 0.2196 0.6905 −0.6571 251 0.0467 0.6902 −0.6979 252 −0.0669 0.7781 −0.5963 253 −0.0075 0.8664 −0.4540 254 0.5639 −0.8218 −0.0813 255 0.4298 −0.8993 −0.0813 256 0.3745 −0.8993 −0.2259 257 0.4745 −0.8218 −0.3154 258 0.5915 −0.7740 −0.2259 259 −0.0669 −0.7781 −0.5963 260 0.0467 −0.6902 −0.6979 261 0.2196 −0.6905 −0.6571 262 0.2800 −0.7856 −0.5196 263 0.1664 −0.8735 −0.4181 264 −0.0075 −0.8664 −0.4540 265 0.6733 −0.5809 −0.4181 266 0.5403 −0.6353 −0.5196 267 0.4881 −0.5354 −0.6571 268 0.5744 −0.3855 −0.6979 269 0.7073 −0.3312 −0.5963 270 0.7540 −0.4267 −0.4540 271 −0.9937 0.0774 −0.0813 272 −0.9660 0.1253 −0.2259 273 −0.9490 0.0000 −0.3154 274 −0.9660 −0.1253 −0.2259 275 −0.9937 −0.0774 −0.0813 276 −0.6405 0.4470 −0.5963 277 −0.6211 0.3047 −0.6979 278 −0.7078 0.1550 −0.6571 279 −0.8203 0.1503 −0.5196 280 −0.8397 0.2926 −0.4181 281 −0.7466 0.4396 −0.4540 282 −0.8397 −0.2926 −0.4181 283 −0.8203 −0.1503 −0.5196 284 −0.7078 −0.1550 −0.6571 285 −0.6211 −0.3047 −0.6979 286 −0.6405 −0.4470 −0.5963 287 −0.7466 −0.4396 −0.4540 288 0.6865 0.0774 −0.7230 289 0.6865 −0.0774 −0.7230 290 0.5694 −0.1253 −0.8124 291 0.4971 0.0000 −0.8677 292 0.5694 0.1253 −0.8124 293 0.0798 −0.4470 −0.8714 294 −0.0023 −0.3047 −0.9342 295 0.0895 −0.1550 −0.9616 296 0.2650 −0.1503 −0.9342 297 0.3472 −0.2926 −0.8714 298 0.2538 −0.4396 −0.8361 299 0.3472 0.2926 −0.8714 300 0.2650 0.1503 −0.9342 301 0.0895 0.1550 −0.9616 302 −0.0023 0.3047 −0.9342 303 0.0798 0.4470 −0.8714 304 0.2538 0.4396 −0.8361 305 −0.4103 0.5558 −0.7230 306 −0.2762 0.6332 −0.7230 307 −0.1762 0.5558 −0.8124 308 −0.2485 0.4305 −0.8677 309 −0.3932 0.4305 −0.8124 310 −0.4270 0.1544 −0.8714 311 −0.2627 0.1544 −0.9342 312 −0.1790 0.0000 −0.9616 313 −0.2627 −0.1544 −0.9342 314 −0.4270 −0.1544 −0.8714 315 −0.5077 0.0000 −0.8361 316 −0.2762 −0.6332 −0.7230 317 −0.4103 −0.5558 −0.7230 318 −0.3932 −0.4305 −0.8124 319 −0.2485 −0.4305 −0.8677 320 −0.1762 −0.5558 −0.8124

TABLE 2 Vertex x/D y/D z/D Face Group of vertices 1 0.0166 0.0382 0.4983 1 96 168 240 216 144 2 0.0166 −0.0382 −0.4983 2 97 169 241 217 145 3 −0.0166 −0.0382 0.4983 3 98 170 242 218 146 4 −0.0166 0.0382 −0.4983 4 99 171 243 219 147 5 0.4983 0.0166 0.0382 5 100 172 244 221 149 6 0.4983 −0.0166 −0.0382 6 101 173 245 220 148 7 −0.4983 −0.0166 0.0382 7 102 174 246 223 151 8 −0.4983 0.0166 −0.0382 8 103 175 247 222 150 9 0.0382 0.4983 0.0166 9 104 176 248 226 154 10 0.0382 −0.4983 −0.0166 10 105 177 249 227 155 11 −0.0382 −0.4983 0.0166 11 106 178 250 224 152 12 −0.0382 0.4983 −0.0166 12 107 179 251 225 153 13 0.0979 0.0465 0.4881 13 72 26 0 12 60 84 14 0.0979 −0.0465 −0.4881 14 72 84 192 264 230 134 15 −0.0979 −0.0465 0.4881 15 72 134 158 110 50 26 16 −0.0979 0.0465 −0.4881 16 73 25 3 15 63 87 17 0.4881 0.0979 0.0465 17 73 87 195 267 229 133 18 0.4881 −0.0979 −0.0465 18 73 133 157 109 49 25 19 −0.4881 −0.0979 0.0465 19 74 24 2 14 62 86 20 −0.4881 0.0979 −0.0465 20 74 86 194 266 228 132 21 0.0465 0.4881 0.0979 21 74 132 156 108 48 24 22 0.0465 −0.4881 −0.0979 22 75 27 1 13 61 85 23 −0.0465 −0.4881 0.0979 23 75 85 193 265 231 135 24 −0.0465 0.4881 −0.0979 24 75 135 159 111 51 27 25 0.0333 −0.1043 0.4879 25 76 28 4 16 64 88 26 0.0333 0.1043 −0.4879 26 76 88 196 268 232 136 27 −0.0333 0.1043 0.4879 27 76 136 160 112 52 28 28 −0.0333 −0.1043 −0.4879 28 77 29 5 17 65 89 29 0.4879 −0.0333 0.1043 29 77 89 197 269 233 137 30 0.4879 0.0333 −0.1043 30 77 137 161 113 53 29 31 −0.4879 0.0333 0.1043 31 78 30 6 18 66 90 32 −0.4879 −0.0333 −0.1043 32 78 90 198 270 234 138 33 0.1043 −0.4879 0.0333 33 78 138 162 114 54 30 34 0.1043 0.4879 −0.0333 34 79 31 7 19 67 91 35 −0.1043 0.4879 0.0333 35 79 91 199 271 235 139 36 −0.1043 −0.4879 −0.0333 36 79 139 163 115 55 31 37 0.1443 −0.0179 0.4784 37 80 33 8 20 68 92 38 0.1443 0.0179 −0.4784 38 80 92 200 272 237 141 39 −0.1443 0.0179 0.4784 39 80 141 165 117 57 33 40 −0.1443 −0.0179 −0.4784 40 81 32 9 21 69 93 41 0.4784 −0.1443 0.0179 41 81 93 201 273 236 140 42 0.4784 0.1443 −0.0179 42 81 140 164 116 56 32 43 −0.4784 0.1443 0.0179 43 82 34 11 23 71 95 44 −0.4784 −0.1443 −0.0179 44 82 95 203 275 238 142 45 0.0179 −0.4784 0.1443 45 82 142 166 118 58 34 46 0.0179 0.4784 −0.1443 46 83 35 10 22 70 94 47 −0.0179 0.4784 0.1443 47 83 94 202 274 239 143 48 −0.0179 −0.4784 −0.1443 48 83 143 167 119 59 35 49 0.1142 −0.0920 0.4780 49 372 360 252 204 312 364 50 0.1142 0.0920 −0.4780 50 372 364 256 208 316 368 51 −0.1142 0.0920 0.4780 51 372 368 260 212 320 360 52 −0.1142 −0.0920 −0.4780 52 373 325 277 349 297 333 53 0.4780 −0.1142 0.0920 53 373 333 285 357 293 329 54 0.4780 0.1142 −0.0920 54 373 329 281 353 289 325 55 −0.4780 0.1142 0.0920 55 374 324 276 348 296 332 56 −0.4780 −0.1142 −0.0920 56 374 332 284 356 292 328 57 0.0920 −0.4780 0.1142 57 374 328 280 352 288 324 58 0.0920 0.4780 −0.1142 58 375 361 253 205 313 365 59 −0.0920 0.4780 0.1142 59 375 365 257 209 317 369 60 −0.0920 −0.4780 −0.1142 60 375 369 261 213 321 361 61 0.1304 0.1181 0.4680 61 376 326 278 350 298 334 62 0.1304 −0.1181 −0.4680 62 376 334 286 358 294 330 63 −0.1304 −0.1181 0.4680 63 376 330 282 354 290 326 64 −0.1304 0.1181 −0.4680 64 377 363 255 207 315 367 65 0.4680 0.1304 0.1181 65 377 367 259 211 319 371 66 0.4680 −0.1304 −0.1181 66 377 371 263 215 323 363 67 −0.4680 −0.1304 0.1181 67 378 362 254 206 314 366 68 −0.4680 0.1304 −0.1181 68 378 366 258 210 318 370 69 0.1181 0.4680 0.1304 69 378 370 262 214 322 362 70 0.1181 −0.4680 −0.1304 70 379 327 279 351 299 335 71 −0.1181 −0.4680 0.1304 71 379 335 287 359 295 331 72 −0.1181 0.4680 −0.1304 72 379 331 283 355 291 327 73 0.0000 0.1784 0.4671 73 48 108 180 168 96 36 74 0.0000 0.1784 −0.4671 74 49 109 181 169 97 37 75 0.0000 −0.1784 0.4671 75 50 110 182 170 98 38 76 0.0000 −0.1784 −0.4671 76 51 111 183 171 99 39 77 0.4671 0.0000 0.1784 77 52 112 184 172 100 40 78 0.4671 0.0000 −0.1784 78 53 113 185 173 101 41 79 −0.4671 0.0000 0.1784 79 54 114 186 174 102 42 80 −0.4671 0.0000 −0.1784 80 55 115 187 175 103 43 81 0.1784 0.4671 0.0000 81 56 116 188 176 104 44 82 0.1784 −0.4671 0.0000 82 57 117 189 177 105 45 83 −0.1784 0.4671 0.0000 83 58 118 190 178 106 46 84 −0.1784 −0.4671 0.0000 84 59 119 191 179 107 47 85 0.0830 0.1847 0.4571 85 60 12 36 96 144 120 86 0.0830 −0.1847 −0.4571 86 61 13 37 97 145 121 87 −0.0830 −0.1847 0.4571 87 62 14 38 98 146 122 88 −0.0830 0.1847 −0.4571 88 63 15 39 99 147 123 89 0.4571 0.0830 0.1847 89 64 16 41 101 148 124 90 0.4571 −0.0830 −0.1847 90 65 17 40 100 149 125 91 −0.4571 −0.0830 0.1847 91 66 18 43 103 150 126 92 −0.4571 0.0830 −0.1847 92 67 19 42 102 151 127 93 0.1847 0.4571 0.0830 93 68 20 46 106 152 128 94 0.1847 −0.4571 −0.0830 94 69 21 47 107 153 129 95 −0.1847 −0.4571 0.0830 95 70 22 44 104 154 130 96 −0.1847 0.4571 −0.0830 96 71 23 45 105 155 131 97 0.2123 −0.0080 0.4526 97 228 266 310 226 248 336 98 0.2123 0.0080 −0.4526 98 229 267 311 227 249 337 99 −0.2123 0.0080 0.4526 99 230 264 308 224 250 338 100 −0.2123 −0.0080 −0.4526 100 231 265 309 225 251 339 101 0.4526 −0.2123 0.0080 101 232 268 300 216 240 340 102 0.4526 0.2123 −0.0080 102 233 269 301 217 241 341 103 −0.4526 0.2123 0.0080 103 234 270 302 218 242 342 104 −0.4526 −0.2123 −0.0080 104 235 271 303 219 243 343 105 0.0080 −0.4526 0.2123 105 236 273 305 221 244 344 106 0.0080 0.4526 −0.2123 106 237 272 304 220 245 345 107 −0.0080 0.4526 0.2123 107 238 275 307 223 246 346 108 −0.0080 −0.4526 −0.2123 108 239 274 306 222 247 347 109 0.1621 −0.1505 0.4484 109 288 352 340 240 168 180 110 0.1621 0.1505 −0.4484 110 289 353 341 241 169 181 111 −0.1621 0.1505 0.4484 111 290 354 342 242 170 182 112 −0.1621 −0.1505 −0.4484 112 291 355 343 243 171 183 113 0.4484 −0.1621 0.1505 113 292 356 344 244 172 184 114 0.4484 0.1621 −0.1505 114 293 357 345 245 173 185 115 −0.4484 0.1621 0.1505 115 294 358 346 246 174 186 116 −0.4484 −0.1621 −0.1505 116 295 359 347 247 175 187 117 0.1505 −0.4484 0.1621 117 296 348 336 248 176 188 118 0.1505 0.4484 −0.1621 118 297 349 337 249 177 189 119 −0.1505 0.4484 0.1621 119 298 350 338 250 178 190 120 −0.1505 −0.4484 −0.1621 120 299 351 339 251 179 191 121 0.2055 0.1169 0.4406 121 312 204 120 144 216 300 122 0.2055 −0.1169 −0.4406 122 313 205 121 145 217 301 123 −0.2055 −0.1169 0.4406 123 314 206 122 146 218 302 124 −0.2055 0.1169 −0.4406 124 315 207 123 147 219 303 125 0.4406 0.2055 0.1169 125 316 208 124 148 220 304 126 0.4406 −0.2055 −0.1169 126 317 209 125 149 221 305 127 −0.4406 −0.2055 0.1169 127 318 210 126 150 222 306 128 −0.4406 0.2055 −0.1169 128 319 211 127 151 223 307 129 0.1169 0.4406 0.2055 129 320 212 128 152 224 308 130 0.1169 −0.4406 −0.2055 130 321 213 129 153 225 309 131 −0.1169 −0.4406 0.2055 131 322 214 130 154 226 310 132 −0.1169 0.4406 −0.2055 132 323 215 131 155 227 311 133 0.0497 −0.2387 0.4365 133 48 36 12 0 2 24 134 0.0497 0.2387 −0.4365 134 49 37 13 1 3 25 135 −0.0497 0.2387 0.4365 135 50 38 14 2 0 26 136 −0.0497 −0.2387 −0.4365 136 51 39 15 3 1 27 137 0.4365 −0.0497 0.2387 137 52 40 17 5 4 28 138 0.4365 0.0497 −0.2387 138 53 41 16 4 5 29 139 −0.4365 0.0497 0.2387 139 54 42 19 7 6 30 140 −0.4365 −0.0497 −0.2387 140 55 43 18 6 7 31 141 0.2387 −0.4365 0.0497 141 56 44 22 10 9 32 142 0.2387 0.4365 −0.0497 142 57 45 23 11 8 33 143 −0.2387 0.4365 0.0497 143 58 46 20 8 11 34 144 −0.2387 −0.4365 −0.0497 144 59 47 21 9 10 35 145 0.2396 0.0521 0.4358 145 60 120 204 252 192 84 146 0.2396 −0.0521 −0.4358 146 61 121 205 253 193 85 147 −0.2396 −0.0521 0.4358 147 62 122 206 254 194 86 148 −0.2396 0.0521 −0.4358 148 63 123 207 255 195 87 149 0.4358 0.2396 0.0521 149 64 124 208 256 196 88 150 0.4358 −0.2396 −0.0521 150 65 125 209 257 197 89 151 −0.4358 −0.2396 0.0521 151 66 126 210 258 198 90 152 −0.4358 0.2396 −0.0521 152 67 127 211 259 199 91 153 0.0521 0.4358 0.2396 153 68 128 212 260 200 92 154 0.0521 −0.4358 −0.2396 154 69 129 213 261 201 93 155 −0.0521 −0.4358 0.2396 155 70 130 214 262 202 94 156 −0.0521 0.4358 −0.2396 156 71 131 215 263 203 95 157 0.1314 −0.2239 0.4274 157 132 228 336 348 276 156 158 0.1314 0.2239 −0.4274 158 133 229 337 349 277 157 159 −0.1314 0.2239 0.4274 159 134 230 338 350 278 158 160 −0.1314 −0.2239 −0.4274 160 135 231 339 351 279 159 161 0.4274 −0.1314 0.2239 161 136 232 340 352 280 160 162 0.4274 0.1314 −0.2239 162 137 233 341 353 281 161 163 −0.4274 0.1314 0.2239 163 138 234 342 354 282 162 164 −0.4274 −0.1314 −0.2239 164 139 235 343 355 283 163 165 0.2239 −0.4274 0.1314 165 140 236 344 356 284 164 166 0.2239 0.4274 −0.1314 166 141 237 345 357 285 165 167 −0.2239 0.4274 0.1314 167 142 238 346 358 286 166 168 −0.2239 −0.4274 −0.1314 168 143 239 347 359 287 167 169 0.2525 −0.0570 0.4278 169 288 180 108 156 276 324 170 0.2525 0.0570 −0.4278 170 289 181 109 157 277 325 171 −0.2525 0.0570 0.4278 171 290 182 110 158 278 326 172 −0.2525 −0.0570 −0.4278 172 291 183 111 159 279 327 173 0.4278 −0.2525 0.0570 173 292 184 112 160 280 328 174 0.4278 0.2525 −0.0570 174 293 185 113 161 281 329 175 −0.4278 0.2525 0.0570 175 294 186 114 162 282 330 176 −0.4278 −0.2525 −0.0570 176 295 187 115 163 283 331 177 0.0570 −0.4278 0.2525 177 296 188 116 164 284 332 178 0.0570 0.4278 −0.2525 178 297 189 117 165 285 333 179 −0.0570 0.4278 0.2525 179 298 190 118 166 286 334 180 −0.0570 −0.4278 −0.2525 180 299 191 119 167 287 335 181 0.2345 −0.1280 0.4227 181 308 264 192 252 360 320 182 0.2345 0.1280 −0.4227 182 309 265 193 253 361 321 183 −0.2345 0.1280 0.4227 183 310 266 194 254 362 322 184 −0.2345 −0.1280 −0.4227 184 311 267 195 255 363 323 185 0.4227 −0.2345 0.1280 185 300 268 196 256 364 312 186 0.4227 0.2345 −0.1280 186 301 269 197 257 365 313 187 −0.4227 0.2345 0.1280 187 302 270 198 258 366 314 188 −0.4227 −0.2345 −0.1280 188 303 271 199 259 367 315 189 0.1280 −0.4227 0.2345 189 304 272 200 260 368 316 190 0.1280 0.4227 −0.2345 190 305 273 201 261 369 317 191 −0.1280 0.4227 0.2345 191 306 274 202 262 370 318 192 −0.1280 −0.4227 −0.2345 192 307 275 203 263 371 319 193 0.1147 0.2508 0.4171 194 0.1147 −0.2508 −0.4171 195 −0.1147 −0.2508 0.4171 196 −0.1147 0.2508 −0.4171 197 0.4171 0.1147 0.2508 198 0.4171 −0.1147 −0.2508 199 −0.4171 −0.1147 0.2508 200 −0.4171 0.1147 −0.2508 201 0.2508 0.4171 0.1147 202 0.2508 −0.4171 −0.1147 203 −0.2508 −0.4171 0.1147 204 −0.2508 0.4171 −0.1147 205 0.2374 0.1793 0.4019 206 0.2374 −0.1793 −0.4019 207 −0.2374 −0.1793 0.4019 208 −0.2374 0.1793 −0.4019 209 0.4019 0.2374 0.1793 210 0.4019 −0.2374 −0.1793 211 −0.4019 −0.2374 0.1793 212 −0.4019 0.2374 −0.1793 213 0.1793 0.4019 0.2374 214 0.1793 −0.4019 −0.2374 215 −0.1793 −0.4019 0.2374 216 −0.1793 0.4019 −0.2374 217 0.2966 0.0401 0.4005 218 0.2966 −0.0401 −0.4005 219 −0.2966 −0.0401 0.4005 220 −0.2966 0.0401 −0.4005 221 0.4005 0.2966 0.0401 222 0.4005 −0.2966 −0.0401 223 −0.4005 −0.2966 0.0401 224 −0.4005 0.2966 −0.0401 225 0.0401 0.4005 0.2966 226 0.0401 −0.4005 −0.2966 227 −0.0401 −0.4005 0.2966 228 −0.0401 0.4005 −0.2966 229 0.0162 −0.3030 0.3974 230 0.0162 0.3030 −0.3974 231 −0.0162 0.3030 0.3974 232 −0.0162 −0.3030 −0.3974 233 0.3974 −0.0162 0.3030 234 0.3974 0.0162 −0.3030 235 −0.3974 0.0162 0.3030 236 −0.3974 −0.0162 −0.3030 237 0.3030 −0.3974 0.0162 238 0.3030 0.3974 −0.0162 239 −0.3030 0.3974 0.0162 240 −0.3030 −0.3974 −0.0162 241 0.3045 −0.0273 0.3956 242 0.3045 0.0273 −0.3956 243 −0.3045 0.0273 0.3956 244 −0.3045 −0.0273 −0.3956 245 0.3956 −0.3045 0.0273 246 0.3956 0.3045 −0.0273 247 −0.3956 0.3045 0.0273 248 −0.3956 −0.3045 −0.0273 249 0.0273 −0.3956 0.3045 250 0.0273 0.3956 −0.3045 251 −0.0273 0.3956 0.3045 252 −0.0273 −0.3956 −0.3045 253 0.1932 0.2475 0.3891 254 0.1932 −0.2475 −0.3891 255 −0.1932 −0.2475 0.3891 256 −0.1932 0.2475 −0.3891 257 0.3891 0.1932 0.2475 258 0.3891 −0.1932 −0.2475 259 −0.3891 −0.1932 0.2475 260 −0.3891 0.1932 −0.2475 261 0.2475 0.3891 0.1932 262 0.2475 −0.3891 −0.1932 263 −0.2475 −0.3891 0.1932 264 −0.2475 0.3891 −0.1932 265 0.0643 0.3089 0.3879 266 0.0643 −0.3089 −0.3879 267 −0.0643 −0.3089 0.3879 268 −0.0643 0.3089 −0.3879 269 0.3879 0.0643 0.3089 270 0.3879 −0.0643 −0.3089 271 −0.3879 −0.0643 0.3089 272 −0.3879 0.0643 −0.3089 273 0.3089 0.3879 0.0643 274 0.3089 −0.3879 −0.0643 275 −0.3089 −0.3879 0.0643 276 −0.3089 0.3879 −0.0643 277 0.1766 −0.2744 0.3788 278 0.1766 0.2744 −0.3788 279 −0.1766 0.2744 0.3788 280 −0.1766 −0.2744 −0.3788 281 0.3788 −0.1766 0.2744 282 0.3788 0.1766 −0.2744 283 −0.3788 0.1766 0.2744 284 −0.3788 −0.1766 −0.2744 285 0.2744 −0.3788 0.1766 286 0.2744 0.3788 −0.1766 287 −0.2744 0.3788 0.1766 288 −0.2744 −0.3788 −0.1766 289 0.2793 −0.1750 0.3760 290 0.2793 0.1750 −0.3760 291 −0.2793 0.1750 0.3760 292 −0.2793 −0.1750 −0.3760 293 0.3760 −0.2793 0.1750 294 0.3760 0.2793 −0.1750 295 −0.3760 0.2793 0.1750 296 −0.3760 −0.2793 −0.1750 297 0.1750 −0.3760 0.2793 298 0.1750 0.3760 −0.2793 299 −0.1750 0.3760 0.2793 300 −0.1750 −0.3760 −0.2793 301 0.3335 0.0901 0.3615 302 0.3335 −0.0901 −0.3615 303 −0.3335 −0.0901 0.3615 304 −0.3335 0.0901 −0.3615 305 0.3615 0.3335 0.0901 306 0.3615 −0.3335 −0.0901 307 −0.3615 −0.3335 0.0901 308 −0.3615 0.3335 −0.0901 309 0.0901 0.3615 0.3335 310 0.0901 −0.3615 −0.3335 311 −0.0901 −0.3615 0.3335 312 −0.0901 0.3615 −0.3335 313 0.3054 0.1650 0.3599 314 0.3054 −0.1650 −0.3599 315 −0.3054 −0.1650 0.3599 316 −0.3054 0.1650 −0.3599 317 0.3599 0.3054 0.1650 318 0.3599 −0.3054 −0.1650 319 −0.3599 −0.3054 0.1650 320 −0.3599 0.3054 −0.1650 321 0.1650 0.3599 0.3054 322 0.1650 −0.3599 −0.3054 323 −0.1650 −0.3599 0.3054 324 −0.1650 0.3599 −0.3054 325 0.2518 −0.2492 0.3528 326 0.2518 0.2492 −0.3528 327 −0.2518 0.2492 0.3528 328 −0.2518 −0.2492 −0.3528 329 0.3528 −0.2518 0.2492 330 0.3528 0.2518 −0.2492 331 −0.3528 0.2518 0.2492 332 −0.3528 −0.2518 −0.2492 333 0.2492 −0.3528 0.2518 334 0.2492 0.3528 −0.2518 335 −0.2492 0.3528 0.2518 336 −0.2492 −0.3528 −0.2518 337 0.0612 −0.3504 0.3514 338 0.0612 0.3504 −0.3514 339 −0.0612 0.3504 0.3514 340 −0.0612 −0.3504 −0.3514 341 0.3514 −0.0612 0.3504 342 0.3514 0.0612 −0.3504 343 −0.3514 0.0612 0.3504 344 −0.3514 −0.0612 −0.3504 345 0.3504 −0.3514 0.0612 346 0.3504 0.3514 −0.0612 347 −0.3504 0.3514 0.0612 348 −0.3504 −0.3514 −0.0612 349 0.1395 −0.3376 0.3414 350 0.1395 0.3376 −0.3414 351 −0.1395 0.3376 0.3414 352 −0.1395 −0.3376 −0.3414 353 0.3414 −0.1395 0.3376 354 0.3414 0.1395 −0.3376 355 −0.3414 0.1395 0.3376 356 −0.3414 −0.1395 −0.3376 357 0.3376 −0.3414 0.1395 358 0.3376 0.3414 −0.1395 359 −0.3376 0.3414 0.1395 360 −0.3376 −0.3414 −0.1395 361 0.2185 0.3031 0.3322 362 0.2185 −0.3031 −0.3322 363 −0.2185 −0.3031 0.3322 364 −0.2185 0.3031 −0.3322 365 0.3322 0.2185 0.3031 366 0.3322 −0.2185 −0.3031 367 −0.3322 −0.2185 0.3031 368 −0.3322 0.2185 −0.3031 369 0.3031 0.3322 0.2185 370 0.3031 −0.3322 −0.2185 371 −0.3031 −0.3322 0.2185 372 −0.3031 0.3322 −0.2185 373 0.2887 0.2887 0.2887 374 0.2887 0.2887 −0.2887 375 0.2887 −0.2887 0.2887 376 0.2887 −0.2887 −0.2887 377 −0.2887 0.2887 0.2887 378 −0.2887 0.2887 −0.2887 379 −0.2887 −0.2887 0.2887 380 −0.2887 −0.2887 −0.2887

TABLE 3 Sphere x/D y/D z/D d/D 1 0.1437 0.0000 −0.7810 0.6114 2 −0.0718 0.1244 −0.7811 0.6115 3 −0.0718 −0.1244 −0.7811 0.6115 4 0.1418 0.2456 −0.7424 0.6127 5 0.1418 −0.2456 −0.7424 0.6127 6 −0.2836 0.0000 −0.7424 0.6127 7 0.3189 0.1091 −0.6648 0.5125 8 −0.0649 0.3307 −0.6648 0.5125 9 0.3189 −0.1091 −0.6648 0.5125 10 −0.2539 0.2216 −0.6648 0.5125 11 −0.0649 −0.3307 −0.6648 0.5125 12 −0.2539 −0.2216 −0.6648 0.5125 13 0.3860 0.0000 −0.5053 0.2902 14 −0.1930 0.3343 −0.5053 0.2902 15 −0.1930 −0.3343 −0.5053 0.2902 16 0.3413 0.3423 −0.6301 0.6115 17 0.1258 0.4668 −0.6301 0.6115 18 0.3413 −0.3423 −0.6301 0.6115 19 0.1258 −0.4668 −0.6301 0.6115 20 −0.4672 0.1244 −0.6301 0.6115 21 −0.4672 −0.1244 −0.6301 0.6115 22 0.4838 0.1766 −0.5388 0.5125 23 −0.0890 0.5073 −0.5388 0.5125 24 0.4838 −0.1766 −0.5388 0.5125 25 −0.3948 0.3307 −0.5388 0.5125 26 −0.0890 −0.5073 −0.5388 0.5125 27 −0.3948 −0.3307 −0.5388 0.5125 28 0.5858 0.0000 −0.4610 0.5125 29 −0.2929 0.5073 −0.4610 0.5125 30 −0.2929 −0.5073 −0.4610 0.5125 31 0.3139 0.5437 −0.4864 0.6115 32 0.3139 −0.5437 −0.4864 0.6115 33 −0.6278 0.0000 −0.4864 0.6114 34 0.5130 0.3974 −0.4589 0.6127 35 0.0876 0.6430 −0.4589 0.6127 36 0.5130 −0.3974 −0.4589 0.6127 37 −0.6006 0.2456 −0.4589 0.6127 38 0.0876 −0.6430 −0.4589 0.6127 39 −0.6006 −0.2456 −0.4589 0.6127 40 0.6611 0.2116 −0.3857 0.6114 41 −0.1473 0.6784 −0.3858 0.6115 42 0.6612 −0.2116 −0.3857 0.6115 43 −0.5138 0.4668 −0.3857 0.6115 44 −0.1473 −0.6784 −0.3857 0.6115 45 −0.5138 −0.4668 −0.3857 0.6115 46 0.4350 0.5351 −0.2829 0.5125 47 0.2460 0.6442 −0.2829 0.5125 48 0.4350 −0.5351 −0.2829 0.5125 49 0.2460 −0.6442 −0.2829 0.5125 50 −0.6809 0.1091 −0.2829 0.5125 51 −0.6809 −0.1091 −0.2829 0.5125 52 0.7424 0.0000 −0.2836 0.6127 53 −0.3712 0.6430 −0.2836 0.6127 54 −0.3712 −0.6430 −0.2836 0.6127 55 0.6337 0.4129 −0.2421 0.6115 56 0.0407 0.7553 −0.2421 0.6115 57 0.6337 −0.4129 −0.2421 0.6115 58 −0.6745 0.3423 −0.2421 0.6115 59 0.0408 −0.7553 −0.2421 0.6115 60 −0.6745 −0.3423 −0.2421 0.6115 61 0.3123 0.5409 −0.1193 0.2902 62 0.3123 −0.5409 −0.1193 0.2902 63 −0.6246 0.0000 −0.1193 0.2902 64 0.7500 0.2116 −0.1533 0.6115 65 0.7500 −0.2116 −0.1533 0.6115 66 −0.1917 0.7553 −0.1533 0.6115 67 −0.5582 0.5437 −0.1533 0.6115 68 −0.1917 −0.7553 −0.1533 0.6115 69 −0.5582 −0.5437 −0.1533 0.6115 70 0.5128 0.5351 −0.0791 0.5125 71 0.2070 0.7117 −0.0791 0.5125 72 0.5128 −0.5351 −0.0791 0.5125 73 0.2070 −0.7117 −0.0791 0.5125 74 −0.7198 0.1766 −0.0791 0.5125 75 −0.7198 −0.1766 −0.0791 0.5125 76 0.7439 0.0000 −0.0469 0.5125 77 −0.3720 0.6442 −0.0469 0.5125 78 −0.3720 −0.6442 −0.0469 0.5125 79 0.6883 0.3974 0.0000 0.6127 80 0.0000 0.7948 0.0000 0.6127 81 0.6883 −0.3974 0.0000 0.6127 82 −0.6883 0.3974 0.0000 0.6127 83 0.0000 −0.7948 0.0000 0.6127 84 −0.6883 −0.3974 0.0000 0.6127 85 0.3720 0.6442 0.0469 0.5125 86 0.3720 −0.6442 0.0469 0.5125 87 −0.7439 0.0000 0.0469 0.5125 88 0.7198 0.1766 0.0791 0.5125 89 0.7198 −0.1766 0.0791 0.5125 90 −0.2070 0.7117 0.0791 0.5125 91 −0.5128 0.5351 0.0791 0.5125 92 −0.2070 −0.7117 0.0791 0.5125 93 −0.5128 −0.5351 0.0791 0.5125 94 0.5582 0.5437 0.1533 0.6115 95 0.1917 0.7553 0.1533 0.6115 96 0.5582 −0.5437 0.1533 0.6115 97 0.1917 −0.7553 0.1533 0.6115 98 −0.7500 0.2116 0.1533 0.6115 99 −0.7500 −0.2116 0.1533 0.6115 100 0.6246 0.0000 0.1193 0.2902 101 −0.3123 0.5409 0.1193 0.2902 102 −0.3123 −0.5409 0.1193 0.2902 103 0.6745 0.3423 0.2421 0.6115 104 −0.0408 0.7553 0.2421 0.6115 105 0.6745 −0.3423 0.2421 0.6115 106 −0.6337 0.4129 0.2421 0.6115 107 −0.0407 −0.7553 0.2421 0.6115 108 −0.6337 −0.4129 0.2421 0.6115 109 0.3712 0.6430 0.2836 0.6127 110 0.3712 −0.6430 0.2836 0.6127 111 −0.7424 0.0000 0.2836 0.6127 112 0.6809 0.1091 0.2829 0.5125 113 0.6809 −0.1091 0.2829 0.5125 114 −0.2460 0.6442 0.2829 0.5125 115 −0.4350 0.5351 0.2829 0.5125 116 −0.2460 −0.6442 0.2829 0.5125 117 −0.4350 −0.5351 0.2829 0.5125 118 0.5138 0.4668 0.3857 0.6115 119 0.1473 0.6784 0.3857 0.6115 120 0.5138 −0.4668 0.3857 0.6115 121 −0.6612 0.2116 0.3857 0.6115 122 0.1473 −0.6784 0.3858 0.6115 123 −0.6611 −0.2116 0.3857 0.6114 124 0.6006 0.2456 0.4589 0.6127 125 −0.0876 0.6430 0.4589 0.6127 126 0.6006 −0.2456 0.4589 0.6127 127 −0.5130 0.3974 0.4589 0.6127 128 −0.0876 −0.6430 0.4589 0.6127 129 −0.5130 −0.3974 0.4589 0.6127 130 0.6278 0.0000 0.4864 0.6114 131 −0.3139 0.5437 0.4864 0.6115 132 −0.3139 −0.5437 0.4864 0.6115 133 0.2929 0.5073 0.4610 0.5125 134 0.2929 −0.5073 0.4610 0.5125 135 −0.5858 0.0000 0.4610 0.5125 136 0.3948 0.3307 0.5388 0.5125 137 0.0890 0.5073 0.5388 0.5125 138 0.3948 −0.3307 0.5388 0.5125 139 −0.4838 0.1766 0.5388 0.5125 140 0.0890 −0.5073 0.5388 0.5125 141 −0.4838 −0.1766 0.5388 0.5125 142 0.4672 0.1244 0.6301 0.6115 143 0.4672 −0.1244 0.6301 0.6115 144 −0.1258 0.4668 0.6301 0.6115 145 −0.3413 0.3423 0.6301 0.6115 146 −0.1258 −0.4668 0.6301 0.6115 147 −0.3413 −0.3423 0.6301 0.6115 148 0.1930 0.3343 0.5053 0.2902 149 0.1930 −0.3343 0.5053 0.2902 150 −0.3860 0.0000 0.5053 0.2902 151 0.2539 0.2216 0.6648 0.5125 152 0.0649 0.3307 0.6648 0.5125 153 0.2539 −0.2216 0.6648 0.5125 154 −0.3189 0.1091 0.6648 0.5125 155 0.0649 −0.3307 0.6648 0.5125 156 −0.3189 −0.1091 0.6648 0.5125 157 0.2836 0.0000 0.7424 0.6127 158 −0.1418 0.2456 0.7424 0.6127 159 −0.1418 −0.2456 0.7424 0.6127 160 0.0718 0.1244 0.7811 0.6115 161 0.0718 −0.1244 0.7811 0.6115 162 −0.1437 0.0000 0.7810 0.6114

TABLE 4 Vertex x/D y/D z/D Face Group of vertices 1 0.0304 0.0117 0.4989 1 204 276 384 312 216 2 0.0304 −0.0117 −0.4989 2 205 277 385 313 217 3 −0.0304 −0.0117 0.4989 3 206 278 386 314 218 4 −0.0304 0.0117 −0.4989 4 207 279 387 315 219 5 0.4989 0.0304 0.0117 5 208 280 388 317 221 6 0.4989 −0.0304 −0.0117 6 209 281 389 316 220 7 −0.4989 −0.0304 0.0117 7 210 282 390 319 223 8 −0.4989 0.0304 −0.0117 8 211 283 391 318 222 9 0.0117 0.4989 0.0304 9 212 284 392 322 226 10 0.0117 −0.4989 −0.0304 10 213 285 393 323 227 11 −0.0117 −0.4989 0.0304 11 214 286 394 320 224 12 −0.0117 0.4989 −0.0304 12 215 287 395 321 225 13 0.0804 −0.0287 0.4926 13 108 50 24 72 132 156 14 0.0804 0.0287 −0.4926 14 108 156 264 360 302 194 15 −0.0804 0.0287 0.4926 15 108 194 182 98 38 50 16 −0.0804 −0.0287 −0.4926 16 109 49 27 75 135 159 17 0.4926 −0.0804 0.0287 17 109 159 267 363 301 193 18 0.4926 0.0804 −0.0287 18 109 193 181 97 37 49 19 −0.4926 0.0804 0.0287 19 110 48 26 74 134 158 20 −0.4926 −0.0804 −0.0287 20 110 158 266 362 300 192 21 0.0287 −0.4926 0.0804 21 110 192 180 96 36 48 22 0.0287 0.4926 −0.0804 22 111 51 25 73 133 157 23 −0.0287 0.4926 0.0804 23 111 157 265 361 303 195 24 −0.0287 −0.4926 −0.0804 24 111 195 183 99 39 51 25 0.0406 0.0756 0.4926 25 112 52 28 76 136 160 26 0.0406 −0.0756 −0.4926 26 112 160 268 364 304 196 27 −0.0406 −0.0756 0.4926 27 112 196 184 100 40 52 28 −0.0406 0.0756 −0.4926 28 113 53 29 77 137 161 29 0.4926 0.0406 0.0756 29 113 161 269 365 305 197 30 0.4926 −0.0406 −0.0756 30 113 197 185 101 41 53 31 −0.4926 −0.0406 0.0756 31 114 54 30 78 138 162 32 −0.4926 0.0406 −0.0756 32 114 162 270 366 306 198 33 0.0756 0.4926 0.0406 33 114 198 186 102 42 54 34 0.0756 −0.4926 −0.0406 34 115 55 31 79 139 163 35 −0.0756 −0.4926 0.0406 35 115 163 271 367 307 199 36 −0.0756 0.4926 −0.0406 36 115 199 187 103 43 55 37 0.0708 −0.0922 0.4863 37 116 57 32 80 140 164 38 0.0708 0.0922 −0.4863 38 116 164 272 368 309 201 39 −0.0708 0.0922 0.4863 39 116 201 189 105 45 57 40 −0.0708 −0.0922 −0.4863 40 117 56 33 81 141 165 41 0.4863 −0.0708 0.0922 41 117 165 273 369 308 200 42 0.4863 0.0708 −0.0922 42 117 200 188 104 44 56 43 −0.4863 0.0708 0.0922 43 118 58 35 83 143 167 44 −0.4863 −0.0708 −0.0922 44 118 167 275 371 310 202 45 0.0922 −0.4863 0.0708 45 118 202 190 106 46 58 46 0.0922 0.4863 −0.0708 46 119 59 34 82 142 166 47 −0.0922 0.4863 0.0708 47 119 166 274 370 311 203 48 −0.0922 −0.4863 −0.0708 48 119 203 191 107 47 59 49 0.0102 −0.1163 0.4862 49 612 588 456 468 600 592 50 0.0102 0.1163 −0.4862 50 612 592 460 472 604 596 51 −0.0102 0.1163 0.4862 51 612 596 464 476 608 588 52 −0.0102 −0.1163 −0.4862 52 613 577 565 549 489 585 53 0.4862 −0.0102 0.1163 53 613 585 573 545 485 581 54 0.4862 0.0102 −0.1163 54 613 581 569 541 481 577 55 −0.4862 0.0102 0.1163 55 614 576 564 548 488 584 56 −0.4862 −0.0102 −0.1163 56 614 584 572 544 484 580 57 0.1163 −0.4862 0.0102 57 614 580 568 540 480 576 58 0.1163 0.4862 −0.0102 58 615 589 457 469 601 593 59 −0.1163 0.4862 0.0102 59 615 593 461 473 605 597 60 −0.1163 −0.4862 −0.0102 60 615 597 465 477 609 589 61 0.1376 −0.0055 0.4807 61 616 578 566 550 490 586 62 0.1376 0.0055 −0.4807 62 616 586 574 546 486 582 63 −0.1376 0.0055 0.4807 63 616 582 570 542 482 578 64 −0.1376 −0.0055 −0.4807 64 617 591 459 471 603 595 65 0.4807 −0.1376 0.0055 65 617 595 463 475 607 599 66 0.4807 0.1376 −0.0055 66 617 599 467 479 611 591 67 −0.4807 0.1376 0.0055 67 618 590 458 470 602 594 68 −0.4807 −0.1376 −0.0055 68 618 594 462 474 606 598 69 0.0055 −0.4807 0.1376 69 618 598 466 478 610 590 70 0.0055 0.4807 −0.1376 70 619 579 567 551 491 587 71 −0.0055 0.4807 0.1376 71 619 587 575 547 487 583 72 −0.0055 −0.4807 −0.1376 72 619 583 571 543 483 579 73 0.1006 0.0965 0.4802 73 12 0 2 26 48 36 74 0.1006 −0.0965 −0.4802 74 13 1 3 27 49 37 75 −0.1006 −0.0965 0.4802 75 14 2 0 24 50 38 76 −0.1006 0.0965 −0.4802 76 15 3 1 25 51 39 77 0.4802 0.1006 0.0965 77 16 5 4 28 52 40 78 0.4802 −0.1006 −0.0965 78 17 4 5 29 53 41 79 −0.4802 −0.1006 0.0965 79 18 7 6 30 54 42 80 −0.4802 0.1006 −0.0965 80 19 6 7 31 55 43 81 0.0965 0.4802 0.1006 81 20 10 9 33 56 44 82 0.0965 −0.4802 −0.1006 82 21 11 8 32 57 45 83 −0.0965 −0.4802 0.1006 83 22 8 11 35 58 46 84 −0.0965 0.4802 −0.1006 84 23 9 10 34 59 47 85 0.1473 0.0548 0.4747 85 132 228 348 396 264 156 86 0.1473 −0.0548 −0.4747 86 133 229 349 397 265 157 87 −0.1473 −0.0548 0.4747 87 134 230 350 398 266 158 88 −0.1473 0.0548 −0.4747 88 135 231 351 399 267 159 89 0.4747 0.1473 0.0548 89 136 232 352 400 268 160 90 0.4747 −0.1473 −0.0548 90 137 233 353 401 269 161 91 −0.4747 −0.1473 0.0548 91 138 234 354 402 270 162 92 −0.4747 0.1473 −0.0548 92 139 235 355 403 271 163 93 0.0548 0.4747 0.1473 93 140 236 356 404 272 164 94 0.0548 −0.4747 −0.1473 94 141 237 357 405 273 165 95 −0.0548 −0.4747 0.1473 95 142 238 358 406 274 166 96 −0.0548 0.4747 −0.1473 96 143 239 359 407 275 167 97 0.1206 −0.1289 0.4678 97 180 192 300 432 420 288 98 0.1206 0.1289 −0.4678 98 181 193 301 433 421 289 99 −0.1206 0.1289 0.4678 99 182 194 302 434 422 290 100 −0.1206 −0.1289 −0.4678 100 183 195 303 435 423 291 101 0.4678 −0.1206 0.1289 101 184 196 304 436 424 292 102 0.4678 0.1206 −0.1289 102 185 197 305 437 425 293 103 −0.4678 0.1206 0.1289 103 186 198 306 438 426 294 104 −0.4678 −0.1206 −0.1289 104 187 199 307 439 427 295 105 0.1289 −0.4678 0.1206 105 188 200 308 440 428 296 106 0.1289 0.4678 −0.1206 106 189 201 309 441 429 297 107 −0.1289 0.4678 0.1206 107 190 202 310 442 430 298 108 −0.1289 −0.4678 −0.1206 108 191 203 311 443 431 299 109 0.0000 0.1784 0.4671 109 336 288 420 564 576 480 110 0.0000 0.1784 −0.4671 110 337 289 421 565 577 481 111 0.0000 −0.1784 0.4671 111 338 290 422 566 578 482 112 0.0000 −0.1784 −0.4671 112 339 291 423 567 579 483 113 0.4671 0.0000 0.1784 113 340 292 424 568 580 484 114 0.4671 0.0000 −0.1784 114 341 293 425 569 581 485 115 −0.4671 0.0000 0.1784 115 342 294 426 570 582 486 116 −0.4671 0.0000 −0.1784 116 343 295 427 571 583 487 117 0.1784 0.4671 0.0000 117 344 296 428 572 584 488 118 0.1784 −0.4671 0.0000 118 345 297 429 573 585 489 119 −0.1784 0.4671 0.0000 119 346 298 430 574 586 490 120 −0.1784 −0.4671 0.0000 120 347 299 431 575 587 491 121 0.1827 −0.0417 0.4636 121 456 588 608 516 396 348 122 0.1827 0.0417 −0.4636 122 457 589 609 517 397 349 123 −0.1827 0.0417 0.4636 123 458 590 610 518 398 350 124 −0.1827 −0.0417 −0.4636 124 459 591 611 519 399 351 125 0.4636 −0.1827 0.0417 125 460 592 600 520 400 352 126 0.4636 0.1827 −0.0417 126 461 593 601 521 401 353 127 −0.4636 0.1827 0.0417 127 462 594 602 522 402 354 128 −0.4636 −0.1827 −0.0417 128 463 595 603 523 403 355 129 0.0417 −0.4636 0.1827 129 464 596 604 524 404 356 130 0.0417 0.4636 −0.1827 130 465 597 605 525 405 357 131 −0.0417 0.4636 0.1827 131 466 598 606 526 406 358 132 −0.0417 −0.4636 −0.1827 132 467 599 607 527 407 359 133 0.1111 0.1573 0.4614 133 12 60 84 72 24 0 134 0.1111 −0.1573 −0.4614 134 13 61 85 73 25 1 135 −0.1111 −0.1573 0.4614 135 14 62 86 74 26 2 136 −0.1111 0.1573 −0.4614 136 15 63 87 75 27 3 137 0.4614 0.1111 0.1573 137 16 64 89 77 29 5 138 0.4614 −0.1111 −0.1573 138 17 65 88 76 28 4 139 −0.4614 −0.1111 0.1573 139 18 66 91 79 31 7 140 −0.4614 0.1111 −0.1573 140 19 67 90 78 30 6 141 0.1573 0.4614 0.1111 141 20 68 94 82 34 10 142 0.1573 −0.4614 −0.1111 142 21 69 95 83 35 11 143 −0.1573 −0.4614 0.1111 143 22 70 92 80 32 8 144 −0.1573 0.4614 −0.1111 144 23 71 93 81 33 9 145 0.1760 −0.1012 0.4569 145 96 180 288 336 252 144 146 0.1760 0.1012 −0.4569 146 97 181 289 337 253 145 147 −0.1760 0.1012 0.4569 147 98 182 290 338 254 146 148 −0.1760 −0.1012 −0.4569 148 99 183 291 339 255 147 149 0.4569 −0.1760 0.1012 149 100 184 292 340 256 148 150 0.4569 0.1760 −0.1012 150 101 185 293 341 257 149 151 −0.4569 0.1760 0.1012 151 102 186 294 342 258 150 152 −0.4569 −0.1760 −0.1012 152 103 187 295 343 259 151 153 0.1012 −0.4569 0.1760 153 104 188 296 344 260 152 154 0.1012 0.4569 −0.1760 154 105 189 297 345 261 153 155 −0.1012 0.4569 0.1760 155 106 190 298 346 262 154 156 −0.1012 −0.4569 −0.1760 156 107 191 299 347 263 155 157 0.0612 0.1989 0.4546 157 228 240 372 468 456 348 158 0.0612 −0.1989 −0.4546 158 229 241 373 469 457 349 159 −0.0612 −0.1989 0.4546 159 230 242 374 470 458 350 160 −0.0612 0.1989 −0.4546 160 231 243 375 471 459 351 161 0.4546 0.0612 0.1989 161 232 244 376 472 460 352 162 0.4546 −0.0612 −0.1989 162 233 245 377 473 461 353 163 −0.4546 −0.0612 0.1989 163 234 246 378 474 462 354 164 −0.4546 0.0612 −0.1989 164 235 247 379 475 463 355 165 0.1989 0.4546 0.0612 165 236 248 380 476 464 356 166 0.1989 −0.4546 −0.0612 166 237 249 381 477 465 357 167 −0.1989 −0.4546 0.0612 167 238 250 382 478 466 358 168 −0.1989 0.4546 −0.0612 168 239 251 383 479 467 359 169 0.2009 0.0711 0.4523 169 264 396 516 536 504 360 170 0.2009 −0.0711 −0.4523 170 265 397 517 537 505 361 171 −0.2009 −0.0711 0.4523 171 266 398 518 538 506 362 172 −0.2009 0.0711 −0.4523 172 267 399 519 539 507 363 173 0.4523 0.2009 0.0711 173 268 400 520 528 508 364 174 0.4523 −0.2009 −0.0711 174 269 401 521 529 509 365 175 −0.4523 −0.2009 0.0711 175 270 402 522 530 510 366 176 −0.4523 0.2009 −0.0711 176 271 403 523 531 511 367 177 0.0711 0.4523 0.2009 177 272 404 524 532 512 368 178 0.0711 −0.4523 −0.2009 178 273 405 525 533 513 369 179 −0.0711 −0.4523 0.2009 179 274 406 526 534 514 370 180 −0.0711 0.4523 −0.2009 180 275 407 527 535 515 371 181 0.1113 −0.1901 0.4489 181 432 560 452 548 564 420 182 0.1113 0.1901 −0.4489 182 433 561 453 549 565 421 183 −0.1113 0.1901 0.4489 183 434 562 454 550 566 422 184 −0.1113 −0.1901 −0.4489 184 435 563 455 551 567 423 185 0.4489 −0.1113 0.1901 185 436 552 444 540 568 424 186 0.4489 0.1113 −0.1901 186 437 553 445 541 569 425 187 −0.4489 0.1113 0.1901 187 438 554 446 542 570 426 188 −0.4489 −0.1113 −0.1901 188 439 555 447 543 571 427 189 0.1901 −0.4489 0.1113 189 440 556 448 544 572 428 190 0.1901 0.4489 −0.1113 190 441 557 449 545 573 429 191 −0.1901 0.4489 0.1113 191 442 558 450 546 574 430 192 −0.1901 −0.4489 −0.1113 192 443 559 451 547 575 431 193 0.0510 −0.2154 0.4483 193 12 36 96 144 120 60 194 0.0510 0.2154 −0.4483 194 13 37 97 145 121 61 195 −0.0510 0.2154 0.4483 195 14 38 98 146 122 62 196 −0.0510 −0.2154 −0.4483 196 15 39 99 147 123 63 197 0.4483 −0.0510 0.2154 197 16 40 100 148 124 64 198 0.4483 0.0510 −0.2154 198 17 41 101 149 125 65 199 −0.4483 0.0510 0.2154 199 18 42 102 150 126 66 200 −0.4483 −0.0510 −0.2154 200 19 43 103 151 127 67 201 0.2154 −0.4483 0.0510 201 20 44 104 152 128 68 202 0.2154 0.4483 −0.0510 202 21 45 105 153 129 69 203 −0.2154 0.4483 0.0510 203 22 46 106 154 130 70 204 −0.2154 −0.4483 −0.0510 204 23 47 107 155 131 71 205 0.2287 −0.0184 0.4443 205 72 84 168 240 228 132 206 0.2287 0.0184 −0.4443 206 73 85 169 241 229 133 207 −0.2287 0.0184 0.4443 207 74 86 170 242 230 134 208 −0.2287 −0.0184 −0.4443 208 75 87 171 243 231 135 209 0.4443 −0.2287 0.0184 209 76 88 172 244 232 136 210 0.4443 0.2287 −0.0184 210 77 89 173 245 233 137 211 −0.4443 0.2287 0.0184 211 78 90 174 246 234 138 212 −0.4443 −0.2287 −0.0184 212 79 91 175 247 235 139 213 0.0184 −0.4443 0.2287 213 80 92 176 248 236 140 214 0.0184 0.4443 −0.2287 214 81 93 177 249 237 141 215 −0.0184 0.4443 0.2287 215 82 94 178 250 238 142 216 −0.0184 −0.4443 −0.2287 216 83 95 179 251 239 143 217 0.2367 0.0315 0.4393 217 252 336 480 540 444 324 218 0.2367 −0.0315 −0.4393 218 253 337 481 541 445 325 219 −0.2367 −0.0315 0.4393 219 254 338 482 542 446 326 220 −0.2367 0.0315 −0.4393 220 255 339 483 543 447 327 221 0.4393 0.2367 0.0315 221 256 340 484 544 448 328 222 0.4393 −0.2367 −0.0315 222 257 341 485 545 449 329 223 −0.4393 −0.2367 0.0315 223 258 342 486 546 450 330 224 −0.4393 0.2367 −0.0315 224 259 343 487 547 451 331 225 0.0315 0.4393 0.2367 225 260 344 488 548 452 332 226 0.0315 −0.4393 −0.2367 226 261 345 489 549 453 333 227 −0.0315 −0.4393 0.2367 227 262 346 490 550 454 334 228 −0.0315 0.4393 −0.2367 228 263 347 491 551 455 335 229 0.1692 0.1724 0.4378 229 300 362 506 500 560 432 230 0.1692 −0.1724 −0.4378 230 301 363 507 501 561 433 231 −0.1692 −0.1724 0.4378 231 302 360 504 502 562 434 232 −0.1692 0.1724 −0.4378 232 303 361 505 503 563 435 233 0.4378 0.1692 0.1724 233 304 364 508 492 552 436 234 0.4378 −0.1692 −0.1724 234 305 365 509 493 553 437 235 −0.4378 −0.1692 0.1724 235 306 366 510 494 554 438 236 −0.4378 0.1692 −0.1724 236 307 367 511 495 555 439 237 0.1724 0.4378 0.1692 237 308 369 513 496 556 440 238 0.1724 −0.4378 −0.1692 238 309 368 512 497 557 441 239 −0.1724 −0.4378 0.1692 239 310 371 515 498 558 442 240 −0.1724 0.4378 −0.1692 240 311 370 514 499 559 443 241 0.2129 0.1273 0.4341 241 372 408 528 520 600 468 242 0.2129 −0.1273 −0.4341 242 373 409 529 521 601 469 243 −0.2129 −0.1273 0.4341 243 374 410 530 522 602 470 244 −0.2129 0.1273 −0.4341 244 375 411 531 523 603 471 245 0.4341 0.2129 0.1273 245 376 412 532 524 604 472 246 0.4341 −0.2129 −0.1273 246 377 413 533 525 605 473 247 −0.4341 −0.2129 0.1273 247 378 414 534 526 606 474 248 −0.4341 0.2129 −0.1273 248 379 415 535 527 607 475 249 0.1273 0.4341 0.2129 249 380 416 536 516 608 476 250 0.1273 −0.4341 −0.2129 250 381 417 537 517 609 477 251 −0.1273 −0.4341 0.2129 251 382 418 538 518 610 478 252 −0.1273 0.4341 −0.2129 252 383 419 539 519 611 479 253 0.2218 −0.1307 0.4286 253 60 120 204 216 168 84 254 0.2218 0.1307 −0.4286 254 61 121 205 217 169 85 255 −0.2218 0.1307 0.4286 255 62 122 206 218 170 86 256 −0.2218 −0.1307 −0.4286 256 63 123 207 219 171 87 257 0.4286 −0.2218 0.1307 257 64 124 208 221 173 89 258 0.4286 0.2218 −0.1307 258 65 125 209 220 172 88 259 −0.4286 0.2218 0.1307 259 66 126 210 223 175 91 260 −0.4286 −0.2218 −0.1307 260 67 127 211 222 174 90 261 0.1307 −0.4286 0.2218 261 68 128 212 226 178 94 262 0.1307 0.4286 −0.2218 262 69 129 213 227 179 95 263 −0.1307 0.4286 0.2218 263 70 130 214 224 176 92 264 −0.1307 −0.4286 −0.2218 264 71 131 215 225 177 93 265 0.0707 0.2558 0.4238 265 144 252 324 276 204 120 266 0.0707 −0.2558 −0.4238 266 145 253 325 277 205 121 267 −0.0707 −0.2558 0.4238 267 146 254 326 278 206 122 268 −0.0707 0.2558 −0.4238 268 147 255 327 279 207 123 269 0.4238 0.0707 0.2558 269 148 256 328 280 208 124 270 0.4238 −0.0707 −0.2558 270 149 257 329 281 209 125 271 −0.4238 −0.0707 0.2558 271 150 258 330 282 210 126 272 −0.4238 0.0707 −0.2558 272 151 259 331 283 211 127 273 0.2558 0.4238 0.0707 273 152 260 332 284 212 128 274 0.2558 −0.4238 −0.0707 274 153 261 333 285 213 129 275 −0.2558 −0.4238 0.0707 275 154 262 334 286 214 130 276 −0.2558 0.4238 −0.0707 276 155 263 335 287 215 131 277 0.2665 −0.0428 0.4209 277 240 168 216 312 408 372 278 0.2665 0.0428 −0.4209 278 241 169 217 313 409 373 279 −0.2665 0.0428 0.4209 279 242 170 218 314 410 374 280 −0.2665 −0.0428 −0.4209 280 243 171 219 315 411 375 281 0.4209 −0.2665 0.0428 281 244 172 220 316 412 376 282 0.4209 0.2665 −0.0428 282 245 173 221 317 413 377 283 −0.4209 0.2665 0.0428 283 246 174 222 318 414 378 284 −0.4209 −0.2665 −0.0428 284 247 175 223 319 415 379 285 0.0428 −0.4209 0.2665 285 248 176 224 320 416 380 286 0.0428 0.4209 −0.2665 286 249 177 225 321 417 381 287 −0.0428 0.4209 0.2665 287 250 178 226 322 418 382 288 −0.0428 −0.4209 −0.2665 288 251 179 227 323 419 383 289 0.1599 −0.2213 0.4189 289 444 552 492 384 276 324 290 0.1599 0.2213 −0.4189 290 445 553 493 385 277 325 291 −0.1599 0.2213 0.4189 291 446 554 494 386 278 326 292 −0.1599 −0.2213 −0.4189 292 447 555 495 387 279 327 293 0.4189 −0.1599 0.2213 293 448 556 496 388 280 328 294 0.4189 0.1599 −0.2213 294 449 557 497 389 281 329 295 −0.4189 0.1599 0.2213 295 450 558 498 390 282 330 296 −0.4189 −0.1599 −0.2213 296 451 559 499 391 283 331 297 0.2213 −0.4189 0.1599 297 452 560 500 392 284 332 298 0.2213 0.4189 −0.1599 298 453 561 501 393 285 333 299 −0.2213 0.4189 0.1599 299 454 562 502 394 286 334 300 −0.2213 −0.4189 −0.1599 300 455 563 503 395 287 335 301 0.0403 −0.2720 0.4176 301 504 536 416 320 394 502 302 0.0403 0.2720 −0.4176 302 505 537 417 321 395 503 303 −0.0403 0.2720 0.4176 303 506 538 418 322 392 500 304 −0.0403 −0.2720 −0.4176 304 507 539 419 323 393 501 305 0.4176 −0.0403 0.2720 305 508 528 408 312 384 492 306 0.4176 0.0403 −0.2720 306 509 529 409 313 385 493 307 −0.4176 0.0403 0.2720 307 510 530 410 314 386 494 308 −0.4176 −0.0403 −0.2720 308 511 531 411 315 387 495 309 0.2720 −0.4176 0.0403 309 512 532 412 316 389 497 310 0.2720 0.4176 −0.0403 310 513 533 413 317 388 496 311 −0.2720 0.4176 0.0403 311 514 534 414 318 391 499 312 −0.2720 −0.4176 −0.0403 312 515 535 415 319 390 498 313 0.2795 0.0379 0.4128 314 0.2795 −0.0379 −0.4128 315 −0.2795 −0.0379 0.4128 316 −0.2795 0.0379 −0.4128 317 0.4128 0.2795 0.0379 318 0.4128 −0.2795 −0.0379 319 −0.4128 −0.2795 0.0379 320 −0.4128 0.2795 −0.0379 321 0.0379 0.4128 0.2795 322 0.0379 −0.4128 −0.2795 323 −0.0379 −0.4128 0.2795 324 −0.0379 0.4128 −0.2795 325 0.2683 −0.0968 0.4106 326 0.2683 0.0968 −0.4106 327 −0.2683 0.0968 0.4106 328 −0.2683 −0.0968 −0.4106 329 0.4106 −0.2683 0.0968 330 0.4106 0.2683 −0.0968 331 −0.4106 0.2683 0.0968 332 −0.4106 −0.2683 −0.0968 333 0.0968 −0.4106 0.2683 334 0.0968 0.4106 −0.2683 335 −0.0968 0.4106 0.2683 336 −0.0968 −0.4106 −0.2683 337 0.2157 −0.1901 0.4091 338 0.2157 0.1901 −0.4091 339 −0.2157 0.1901 0.4091 340 −0.2157 −0.1901 −0.4091 341 0.4091 −0.2157 0.1901 342 0.4091 0.2157 −0.1901 343 −0.4091 0.2157 0.1901 344 −0.4091 −0.2157 −0.1901 345 0.1901 −0.4091 0.2157 346 0.1901 0.4091 −0.2157 347 −0.1901 0.4091 0.2157 348 −0.1901 −0.4091 −0.2157 349 0.1788 0.2285 0.4072 350 0.1788 −0.2285 −0.4072 351 −0.1788 −0.2285 0.4072 352 −0.1788 0.2285 −0.4072 353 0.4072 0.1788 0.2285 354 0.4072 −0.1788 −0.2285 355 −0.4072 −0.1788 0.2285 356 −0.4072 0.1788 −0.2285 357 0.2285 0.4072 0.1788 358 0.2285 −0.4072 −0.1788 359 −0.2285 −0.4072 0.1788 360 −0.2285 0.4072 −0.1788 361 0.0200 0.2917 0.4056 362 0.0200 −0.2917 −0.4056 363 −0.0200 −0.2917 0.4056 364 −0.0200 0.2917 −0.4056 365 0.4056 0.0200 0.2917 366 0.4056 −0.0200 −0.2917 367 −0.4056 −0.0200 0.2917 368 −0.4056 0.0200 −0.2917 369 0.2917 0.4056 0.0200 370 0.2917 −0.4056 −0.0200 371 −0.2917 −0.4056 0.0200 372 −0.2917 0.4056 −0.0200 373 0.2647 0.1351 0.4021 374 0.2647 −0.1351 −0.4021 375 −0.2647 −0.1351 0.4021 376 −0.2647 0.1351 −0.4021 377 0.4021 0.2647 0.1351 378 0.4021 −0.2647 −0.1351 379 −0.4021 −0.2647 0.1351 380 −0.4021 0.2647 −0.1351 381 0.1351 0.4021 0.2647 382 0.1351 −0.4021 −0.2647 383 −0.1351 −0.4021 0.2647 384 −0.1351 0.4021 −0.2647 385 0.2980 −0.0080 0.4014 386 0.2980 0.0080 −0.4014 387 −0.2980 0.0080 0.4014 388 −0.2980 −0.0080 −0.4014 389 0.4014 −0.2980 0.0080 390 0.4014 0.2980 −0.0080 391 −0.4014 0.2980 0.0080 392 −0.4014 −0.2980 −0.0080 393 0.0080 −0.4014 0.2980 394 0.0080 0.4014 −0.2980 395 −0.0080 0.4014 0.2980 396 −0.0080 −0.4014 −0.2980 397 0.1296 0.2704 0.4001 398 0.1296 −0.2704 −0.4001 399 −0.1296 −0.2704 0.4001 400 −0.1296 0.2704 −0.4001 401 0.4001 0.1296 0.2704 402 0.4001 −0.1296 −0.2704 403 −0.4001 −0.1296 0.2704 404 −0.4001 0.1296 −0.2704 405 0.2704 0.4001 0.1296 406 0.2704 −0.4001 −0.1296 407 −0.2704 −0.4001 0.1296 408 −0.2704 0.4001 −0.1296 409 0.2977 0.0856 0.3925 410 0.2977 −0.0856 −0.3925 411 −0.2977 −0.0856 0.3925 412 −0.2977 0.0856 −0.3925 413 0.3925 0.2977 0.0856 414 0.3925 −0.2977 −0.0856 415 −0.3925 −0.2977 0.0856 416 −0.3925 0.2977 −0.0856 417 0.0856 0.3925 0.2977 418 0.0856 −0.3925 −0.2977 419 −0.0856 −0.3925 0.2977 420 −0.0856 0.3925 −0.2977 421 0.1484 −0.2776 0.3884 422 0.1484 0.2776 −0.3884 423 −0.1484 0.2776 0.3884 424 −0.1484 −0.2776 −0.3884 425 0.3884 −0.1484 0.2776 426 0.3884 0.1484 −0.2776 427 −0.3884 0.1484 0.2776 428 −0.3884 −0.1484 −0.2776 429 0.2776 −0.3884 0.1484 430 0.2776 0.3884 −0.1484 431 −0.2776 0.3884 0.1484 432 −0.2776 −0.3884 −0.1484 433 0.0888 −0.3025 0.3881 434 0.0888 0.3025 −0.3881 435 −0.0888 0.3025 0.3881 436 −0.0888 −0.3025 −0.3881 437 0.3881 −0.0888 0.3025 438 0.3881 0.0888 −0.3025 439 −0.3881 0.0888 0.3025 440 −0.3881 −0.0888 −0.3025 441 0.3025 −0.3881 0.0888 442 0.3025 0.3881 −0.0888 443 −0.3025 0.3881 0.0888 444 −0.3025 −0.3881 −0.0888 445 0.3111 −0.1174 0.3734 446 0.3111 0.1174 −0.3734 447 −0.3111 0.1174 0.3734 448 −0.3111 −0.1174 −0.3734 449 0.3734 −0.3111 0.1174 450 0.3734 0.3111 −0.1174 451 −0.3734 0.3111 0.1174 452 −0.3734 −0.3111 −0.1174 453 0.1174 −0.3734 0.3111 454 0.1174 0.3734 −0.3111 455 −0.1174 0.3734 0.3111 456 −0.1174 −0.3734 −0.3111 457 0.2337 0.2368 0.3732 458 0.2337 −0.2368 −0.3732 459 −0.2337 −0.2368 0.3732 460 −0.2337 0.2368 −0.3732 461 0.3732 0.2337 0.2368 462 0.3732 −0.2337 −0.2368 463 −0.3732 −0.2337 0.2368 464 −0.3732 0.2337 −0.2368 465 0.2368 0.3732 0.2337 466 0.2368 −0.3732 −0.2337 467 −0.2368 −0.3732 0.2337 468 −0.2368 0.3732 −0.2337 469 0.2768 0.1885 0.3713 470 0.2768 −0.1885 −0.3713 471 −0.2768 −0.1885 0.3713 472 −0.2768 0.1885 −0.3713 473 0.3713 0.2768 0.1885 474 0.3713 −0.2768 −0.1885 475 −0.3713 −0.2768 0.1885 476 −0.3713 0.2768 −0.1885 477 0.1885 0.3713 0.2768 478 0.1885 −0.3713 −0.2768 479 −0.1885 −0.3713 0.2768 480 −0.1885 0.3713 −0.2768 481 0.2603 −0.2143 0.3692 482 0.2603 0.2143 −0.3692 483 −0.2603 0.2143 0.3692 484 −0.2603 −0.2143 −0.3692 485 0.3692 −0.2603 0.2143 486 0.3692 0.2603 −0.2143 487 −0.3692 0.2603 0.2143 488 −0.3692 −0.2603 −0.2143 489 0.2143 −0.3692 0.2603 490 0.2143 0.3692 −0.2603 491 −0.2143 0.3692 0.2603 492 −0.2143 −0.3692 −0.2603 493 0.3394 −0.0182 0.3667 494 0.3394 0.0182 −0.3667 495 −0.3394 0.0182 0.3667 496 −0.3394 −0.0182 −0.3667 497 0.3667 −0.3394 0.0182 498 0.3667 0.3394 −0.0182 499 −0.3667 0.3394 0.0182 500 −0.3667 −0.3394 −0.0182 501 0.0182 −0.3667 0.3394 502 0.0182 0.3667 −0.3394 503 −0.0182 0.3667 0.3394 504 −0.0182 −0.3667 −0.3394 505 0.0287 0.3396 0.3659 506 0.0287 −0.3396 −0.3659 507 −0.0287 −0.3396 0.3659 508 −0.0287 0.3396 −0.3659 509 0.3659 0.0287 0.3396 510 0.3659 −0.0287 −0.3396 511 −0.3659 −0.0287 0.3396 512 −0.3659 0.0287 −0.3396 513 0.3396 0.3659 0.0287 514 0.3396 −0.3659 −0.0287 515 −0.3396 −0.3659 0.0287 516 −0.3396 0.3659 −0.0287 517 0.1352 0.3203 0.3593 518 0.1352 −0.3203 −0.3593 519 −0.1352 −0.3203 0.3593 520 −0.1352 0.3203 −0.3593 521 0.3593 0.1352 0.3203 522 0.3593 −0.1352 −0.3203 523 −0.3593 −0.1352 0.3203 524 −0.3593 0.1352 −0.3203 525 0.3203 0.3593 0.1352 526 0.3203 −0.3593 −0.1352 527 −0.3203 −0.3593 0.1352 528 −0.3203 0.3593 −0.1352 529 0.3436 0.0842 0.3533 530 0.3436 −0.0842 −0.3533 531 −0.3436 −0.0842 0.3533 532 −0.3436 0.0842 −0.3533 533 0.3533 0.3436 0.0842 534 0.3533 −0.3436 −0.0842 535 −0.3533 −0.3436 0.0842 536 −0.3533 0.3436 −0.0842 537 0.0842 0.3533 0.3436 538 0.0842 −0.3533 −0.3436 539 −0.0842 −0.3533 0.3436 540 −0.0842 0.3533 −0.3436 541 0.3091 −0.1762 0.3513 542 0.3091 0.1762 −0.3513 543 −0.3091 0.1762 0.3513 544 −0.3091 −0.1762 −0.3513 545 0.3513 −0.3091 0.1762 546 0.3513 0.3091 −0.1762 547 −0.3513 0.3091 0.1762 548 −0.3513 −0.3091 −0.1762 549 0.1762 −0.3513 0.3091 550 0.1762 0.3513 −0.3091 551 −0.1762 0.3513 0.3091 552 −0.1762 −0.3513 −0.3091 553 0.3491 −0.0753 0.3499 554 0.3491 0.0753 −0.3499 555 −0.3491 0.0753 0.3499 556 −0.3491 −0.0753 −0.3499 557 0.3499 −0.3491 0.0753 558 0.3499 0.3491 −0.0753 559 −0.3499 0.3491 0.0753 560 −0.3499 −0.3491 −0.0753 561 0.0753 −0.3499 0.3491 562 0.0753 0.3499 −0.3491 563 −0.0753 0.3499 0.3491 564 −0.0753 −0.3499 −0.3491 565 0.1931 −0.3025 0.3481 566 0.1931 0.3025 −0.3481 567 −0.1931 0.3025 0.3481 568 −0.1931 −0.3025 −0.3481 569 0.3481 −0.1931 0.3025 570 0.3481 0.1931 −0.3025 571 −0.3481 0.1931 0.3025 572 −0.3481 −0.1931 −0.3025 573 0.3025 −0.3481 0.1931 574 0.3025 0.3481 −0.1931 575 −0.3025 0.3481 0.1931 576 −0.3025 −0.3481 −0.1931 577 0.2495 −0.2708 0.3383 578 0.2495 0.2708 −0.3383 579 −0.2495 0.2708 0.3383 580 −0.2495 −0.2708 −0.3383 581 0.3383 −0.2495 0.2708 582 0.3383 0.2495 −0.2708 583 −0.3383 0.2495 0.2708 584 −0.3383 −0.2495 −0.2708 585 0.2708 −0.3383 0.2495 586 0.2708 0.3383 −0.2495 587 −0.2708 0.3383 0.2495 588 −0.2708 −0.3383 −0.2495 589 0.2392 0.2873 0.3320 590 0.2392 −0.2873 −0.3320 591 −0.2392 −0.2873 0.3320 592 −0.2392 0.2873 −0.3320 593 0.3320 0.2392 0.2873 594 0.3320 −0.2392 −0.2873 595 −0.3320 −0.2392 0.2873 596 −0.3320 0.2392 −0.2873 597 0.2873 0.3320 0.2392 598 0.2873 −0.3320 −0.2392 599 −0.2873 −0.3320 0.2392 600 −0.2873 0.3320 −0.2392 601 0.3254 0.1894 0.3289 602 0.3254 −0.1894 −0.3289 603 −0.3254 −0.1894 0.3289 604 −0.3254 0.1894 −0.3289 605 0.3289 0.3254 0.1894 606 0.3289 −0.3254 −0.1894 607 −0.3289 −0.3254 0.1894 608 −0.3289 0.3254 −0.1894 609 0.1894 0.3289 0.3254 610 0.1894 −0.3289 −0.3254 611 −0.1894 −0.3289 0.3254 612 −0.1894 0.3289 −0.3254 613 0.2887 0.2887 0.2887 614 0.2887 0.2887 −0.2887 615 0.2887 −0.2887 0.2887 616 0.2887 −0.2887 −0.2887 617 −0.2887 0.2887 0.2887 618 −0.2887 0.2887 −0.2887 619 −0.2887 −0.2887 0.2887 620 −0.2887 −0.2887 −0.2887

TABLE 5 Sphere x/D y/D z/D d/D 1 0.3189 0.0000 0.5160 0.2254 2 0.3189 0.0000 −0.5160 0.2254 3 −0.3189 0.0000 0.5160 0.2254 4 −0.3189 0.0000 −0.5160 0.2254 5 0.5160 −0.3189 0.0000 0.2254 6 0.5160 0.3189 0.0000 0.2254 7 −0.5160 0.3189 0.0000 0.2254 8 −0.5160 −0.3189 0.0000 0.2254 9 0.0000 −0.5160 0.3189 0.2254 10 0.0000 0.5160 −0.3189 0.2254 11 0.0000 0.5160 0.3189 0.2254 12 0.0000 −0.5160 −0.3189 0.2254 13 0.0794 0.2166 0.7473 0.5800 14 0.0159 0.3710 0.6884 0.5800 15 −0.0954 0.2425 0.7375 0.5800 16 −0.0794 0.2166 −0.7473 0.5800 17 −0.0159 0.3710 −0.6884 0.5800 18 0.0954 0.2425 −0.7375 0.5800 19 −0.0794 −0.2166 0.7473 0.5800 20 −0.0159 −0.3710 0.6884 0.5800 21 0.0954 −0.2425 0.7375 0.5800 22 0.0794 −0.2166 −0.7473 0.5800 23 0.0159 −0.3710 −0.6884 0.5800 24 −0.0954 −0.2425 −0.7375 0.5800 25 0.7473 0.0794 0.2166 0.5800 26 0.6884 0.0159 0.3710 0.5800 27 0.7375 −0.0954 0.2425 0.5800 28 0.7473 −0.0794 −0.2166 0.5800 29 0.6884 −0.0159 −0.3710 0.5800 30 0.7375 0.0954 −0.2425 0.5800 31 −0.7473 −0.0794 0.2166 0.5800 32 −0.6884 −0.0159 0.3710 0.5800 33 −0.7375 0.0954 0.2425 0.5800 34 −0.7473 0.0794 −0.2166 0.5800 35 −0.6884 0.0159 −0.3710 0.5800 36 −0.7375 −0.0954 −0.2425 0.5800 37 0.2166 0.7473 0.0794 0.5800 38 0.3710 0.6884 0.0159 0.5800 39 0.2425 0.7375 −0.0954 0.5800 40 0.2166 −0.7473 −0.0794 0.5800 41 0.3710 −0.6884 −0.0159 0.5800 42 0.2425 −0.7375 0.0954 0.5800 43 −0.2166 0.7473 −0.0794 0.5800 44 −0.3710 0.6884 −0.0159 0.5800 45 −0.2425 0.7375 0.0954 0.5800 46 −0.2166 −0.7473 0.0794 0.5800 47 −0.3710 −0.6884 0.0159 0.5800 48 −0.2425 −0.7375 −0.0954 0.5800 49 0.4459 0.3763 0.5208 0.5800 50 0.5208 0.4459 0.3763 0.5800 51 0.3763 0.5208 0.4459 0.5800 52 0.3665 0.5049 −0.4717 0.5800 53 0.5049 0.4717 −0.3665 0.5800 54 0.4717 0.3665 −0.5049 0.5800 55 0.3665 −0.5049 0.4717 0.5800 56 0.5049 −0.4717 0.3665 0.5800 57 0.4717 −0.3665 0.5049 0.5800 58 0.4459 −0.3763 −0.5208 0.5800 59 0.5208 −0.4459 −0.3763 0.5800 60 0.3763 −0.5208 −0.4459 0.5800 61 −0.3665 0.5049 0.4717 0.5800 62 −0.5049 0.4717 0.3665 0.5800 63 −0.4717 0.3665 0.5049 0.5800 64 −0.4459 0.3763 −0.5208 0.5800 65 −0.5208 0.4459 −0.3763 0.5800 66 −0.3763 0.5208 −0.4459 0.5800 67 −0.4459 −0.3763 0.5208 0.5800 68 −0.5208 −0.4459 0.3763 0.5800 69 −0.3763 −0.5208 0.4459 0.5800 70 −0.3665 −0.5049 −0.4717 0.5800 71 −0.5049 −0.4717 −0.3665 0.5800 72 −0.4717 −0.3665 −0.5049 0.5800 73 0.0315 −0.0822 0.7752 0.5761 74 0.0315 0.0822 −0.7752 0.5761 75 −0.0315 0.0822 0.7752 0.5761 76 −0.0315 −0.0822 −0.7752 0.5761 77 0.7752 −0.0315 0.0822 0.5761 78 0.7752 0.0315 −0.0822 0.5761 79 −0.7752 0.0315 0.0822 0.5761 80 −0.7752 −0.0315 −0.0822 0.5761 81 0.0822 −0.7752 0.0315 0.5761 82 0.0822 0.7752 −0.0315 0.5761 83 −0.0822 0.7752 0.0315 0.5761 84 −0.0822 −0.7752 −0.0315 0.5761 85 0.1888 0.3367 0.6780 0.5761 86 0.1888 −0.3367 −0.6780 0.5761 87 −0.1888 −0.3367 0.6780 0.5761 88 −0.1888 0.3367 −0.6780 0.5761 89 0.6780 0.1888 0.3367 0.5761 90 0.6780 −0.1888 −0.3367 0.5761 91 −0.6780 −0.1888 0.3367 0.5761 92 −0.6780 0.1888 −0.3367 0.5761 93 0.3367 0.6780 0.1888 0.5761 94 0.3367 −0.6780 −0.1888 0.5761 95 −0.3367 −0.6780 0.1888 0.5761 96 −0.3367 0.6780 −0.1888 0.5761 97 0.1573 −0.3877 0.6585 0.5761 98 0.1573 0.3877 −0.6585 0.5761 99 −0.1573 0.3877 0.6585 0.5761 100 −0.1573 −0.3877 −0.6585 0.5761 101 0.6585 −0.1573 0.3877 0.5761 102 0.6585 0.1573 −0.3877 0.5761 103 −0.6585 0.1573 0.3877 0.5761 104 −0.6585 −0.1573 −0.3877 0.5761 105 0.3877 −0.6585 0.1573 0.5761 106 0.3877 0.6585 −0.1573 0.5761 107 −0.3877 0.6585 0.1573 0.5761 108 −0.3877 −0.6585 −0.1573 0.5761 109 0.3218 −0.3875 0.5958 0.5761 110 0.3218 0.3875 −0.5958 0.5761 111 −0.3218 0.3875 0.5958 0.5761 112 −0.3218 −0.3875 −0.5958 0.5761 113 0.5958 −0.3218 0.3875 0.5761 114 0.5958 0.3218 −0.3875 0.5761 115 −0.5958 0.3218 0.3875 0.5761 116 −0.5958 −0.3218 −0.3875 0.5761 117 0.3875 −0.5958 0.3218 0.5761 118 0.3875 0.5958 −0.3218 0.5761 119 −0.3875 0.5958 0.3218 0.5761 120 −0.3875 −0.5958 −0.3218 0.5761 121 0.2903 0.4385 0.5763 0.5761 122 0.2903 −0.4385 −0.5763 0.5761 123 −0.2903 −0.4385 0.5763 0.5761 124 −0.2903 0.4385 −0.5763 0.5761 125 0.5763 0.2903 0.4385 0.5761 126 0.5763 −0.2903 −0.4385 0.5761 127 −0.5763 −0.2903 0.4385 0.5761 128 −0.5763 0.2903 −0.4385 0.5761 129 0.4385 0.5763 0.2903 0.5761 130 0.4385 −0.5763 −0.2903 0.5761 131 −0.4385 −0.5763 0.2903 0.5761 132 −0.4385 0.5763 −0.2903 0.5761 133 0.1373 0.0529 0.7524 0.5487 134 0.1373 −0.0529 −0.7524 0.5487 135 −0.1373 −0.0529 0.7524 0.5487 136 −0.1373 0.0529 −0.7524 0.5487 137 0.7524 −0.1373 −0.0529 0.5487 138 0.7524 0.1373 0.0529 0.5487 139 −0.7524 0.1373 −0.0529 0.5487 140 −0.7524 −0.1373 0.0529 0.5487 141 −0.0529 −0.7524 0.1373 0.5487 142 −0.0529 0.7524 −0.1373 0.5487 143 0.0529 0.7524 0.1373 0.5487 144 0.0529 −0.7524 −0.1373 0.5487 145 0.2584 −0.2488 0.6775 0.5487 146 0.2584 0.2488 −0.6775 0.5487 147 −0.2584 0.2488 0.6775 0.5487 148 −0.2584 −0.2488 −0.6775 0.5487 149 0.6775 −0.2584 0.2488 0.5487 150 0.6775 0.2584 −0.2488 0.5487 151 −0.6775 0.2584 0.2488 0.5487 152 −0.6775 −0.2584 −0.2488 0.5487 153 0.2488 −0.6775 0.2584 0.5487 154 0.2488 0.6775 −0.2584 0.5487 155 −0.2488 0.6775 0.2584 0.5487 156 −0.2488 −0.6775 −0.2584 0.5487 157 0.3439 0.2815 0.6247 0.5487 158 0.3439 −0.2815 −0.6247 0.5487 159 −0.3439 −0.2815 0.6247 0.5487 160 −0.3439 0.2815 −0.6247 0.5487 161 0.6247 0.3439 0.2815 0.5487 162 0.6247 −0.3439 −0.2815 0.5487 163 −0.6247 −0.3439 0.2815 0.5487 164 −0.6247 0.3439 −0.2815 0.5487 165 0.2815 0.6247 0.3439 0.5487 166 0.2815 −0.6247 −0.3439 0.5487 167 −0.2815 −0.6247 0.3439 0.5487 168 −0.2815 0.6247 −0.3439 0.5487 169 0.1211 0.4709 0.5927 0.5487 170 0.1211 −0.4709 −0.5927 0.5487 171 −0.1211 −0.4709 0.5927 0.5487 172 −0.1211 0.4709 −0.5927 0.5487 173 0.5927 0.1211 0.4709 0.5487 174 0.5927 −0.1211 −0.4709 0.5487 175 −0.5927 −0.1211 0.4709 0.5487 176 −0.5927 0.1211 −0.4709 0.5487 177 0.4709 0.5927 0.1211 0.5487 178 0.4709 −0.5927 −0.1211 0.5487 179 −0.4709 −0.5927 0.1211 0.5487 180 −0.4709 0.5927 −0.1211 0.5487 181 0.2066 −0.5036 0.5399 0.5487 182 0.2066 0.5036 −0.5399 0.5487 183 −0.2066 0.5036 0.5399 0.5487 184 −0.2066 −0.5036 −0.5399 0.5487 185 0.5399 −0.2066 0.5036 0.5487 186 0.5399 0.2066 −0.5036 0.5487 187 −0.5399 0.2066 0.5036 0.5487 188 −0.5399 −0.2066 −0.5036 0.5487 189 0.5036 −0.5399 0.2066 0.5487 190 0.5036 0.5399 −0.2066 0.5487 191 −0.5036 0.5399 0.2066 0.5487 192 −0.5036 −0.5399 −0.2066 0.5487 193 0.1942 −0.1019 0.7251 0.5304 194 0.1942 0.1019 −0.7251 0.5304 195 −0.1942 0.1019 0.7251 0.5304 196 −0.1942 −0.1019 −0.7251 0.5304 197 0.7251 −0.1942 0.1019 0.5304 198 0.7251 0.1942 −0.1019 0.5304 199 −0.7251 0.1942 0.1019 0.5304 200 −0.7251 −0.1942 −0.1019 0.5304 201 0.1019 −0.7251 0.1942 0.5304 202 0.1019 0.7251 −0.1942 0.5304 203 −0.1019 0.7251 0.1942 0.5304 204 −0.1019 −0.7251 −0.1942 0.5304 205 0.2387 0.1739 0.6976 0.5304 206 0.2387 −0.1739 −0.6976 0.5304 207 −0.2387 −0.1739 0.6976 0.5304 208 −0.2387 0.1739 −0.6976 0.5304 209 0.6976 0.2387 0.1739 0.5304 210 0.6976 −0.2387 −0.1739 0.5304 211 −0.6976 −0.2387 0.1739 0.5304 212 −0.6976 0.2387 −0.1739 0.5304 213 0.1739 0.6976 0.2387 0.5304 214 0.1739 −0.6976 −0.2387 0.5304 215 −0.1739 −0.6976 0.2387 0.5304 216 −0.1739 0.6976 −0.2387 0.5304 217 0.4037 −0.2369 0.5957 0.5304 218 0.4037 0.2369 −0.5957 0.5304 219 −0.4037 0.2369 0.5957 0.5304 220 −0.4037 −0.2369 −0.5957 0.5304 221 0.5957 −0.4037 0.2369 0.5304 222 0.5957 0.4037 −0.2369 0.5304 223 −0.5957 0.4037 0.2369 0.5304 224 −0.5957 −0.4037 −0.2369 0.5304 225 0.2369 −0.5957 0.4037 0.5304 226 0.2369 0.5957 −0.4037 0.5304 227 −0.2369 0.5957 0.4037 0.5304 228 −0.2369 −0.5957 −0.4037 0.5304 229 0.0445 −0.4882 0.5776 0.5304 230 0.0445 0.4882 −0.5776 0.5304 231 −0.0445 0.4882 0.5776 0.5304 232 −0.0445 −0.4882 −0.5776 0.5304 233 0.5776 −0.0445 0.4882 0.5304 234 0.5776 0.0445 −0.4882 0.5304 235 −0.5776 0.0445 0.4882 0.5304 236 −0.5776 −0.0445 −0.4882 0.5304 237 0.4882 −0.5776 0.0445 0.5304 238 0.4882 0.5776 −0.0445 0.5304 239 −0.4882 0.5776 0.0445 0.5304 240 −0.4882 −0.5776 −0.0445 0.5304 241 0.4757 0.2094 0.5512 0.5304 242 0.4757 −0.2094 −0.5512 0.5304 243 −0.4757 −0.2094 0.5512 0.5304 244 −0.4757 0.2094 −0.5512 0.5304 245 0.5512 0.4757 0.2094 0.5304 246 0.5512 −0.4757 −0.2094 0.5304 247 −0.5512 −0.4757 0.2094 0.5304 248 −0.5512 0.4757 −0.2094 0.5304 249 0.2094 0.5512 0.4757 0.5304 250 0.2094 −0.5512 −0.4757 0.5304 251 −0.2094 −0.5512 0.4757 0.5304 252 −0.2094 0.5512 −0.4757 0.5304 253 0.2679 0.0224 0.6585 0.4368 254 0.2679 −0.0224 −0.6585 0.4368 255 −0.2679 −0.0224 0.6585 0.4368 256 −0.2679 0.0224 −0.6585 0.4368 257 0.6585 −0.2679 −0.0224 0.4368 258 0.6585 0.2679 0.0224 0.4368 259 −0.6585 0.2679 −0.0224 0.4368 260 −0.6585 −0.2679 0.0224 0.4368 261 −0.0224 −0.6585 0.2679 0.4368 262 −0.0224 0.6585 −0.2679 0.4368 263 0.0224 0.6585 0.2679 0.4368 264 0.0224 −0.6585 −0.2679 0.4368 265 0.3193 −0.1056 0.6267 0.4368 266 0.3193 0.1056 −0.6267 0.4368 267 −0.3193 0.1056 0.6267 0.4368 268 −0.3193 −0.1056 −0.6267 0.4368 269 0.6267 −0.3193 0.1056 0.4368 270 0.6267 0.3193 −0.1056 0.4368 271 −0.6267 0.3193 0.1056 0.4368 272 −0.6267 −0.3193 −0.1056 0.4368 273 0.1056 −0.6267 0.3193 0.4368 274 0.1056 0.6267 −0.3193 0.4368 275 −0.1056 0.6267 0.3193 0.4368 276 −0.1056 −0.6267 −0.3193 0.4368 277 0.3556 0.1194 0.6043 0.4368 278 0.3556 −0.1194 −0.6043 0.4368 279 −0.3556 −0.1194 0.6043 0.4368 280 −0.3556 0.1194 −0.6043 0.4368 281 0.6043 0.3556 0.1194 0.4368 282 0.6043 −0.3556 −0.1194 0.4368 283 −0.6043 −0.3556 0.1194 0.4368 284 −0.6043 0.3556 −0.1194 0.4368 285 0.1194 0.6043 0.3556 0.4368 286 0.1194 −0.6043 −0.3556 0.4368 287 −0.1194 −0.6043 0.3556 0.4368 288 −0.1194 0.6043 −0.3556 0.4368 289 0.4387 −0.0877 0.5529 0.4368 290 0.4387 0.0877 −0.5529 0.4368 291 −0.4387 0.0877 0.5529 0.4368 292 −0.4387 −0.0877 −0.5529 0.4368 293 0.5529 −0.4387 0.0877 0.4368 294 0.5529 0.4387 −0.0877 0.4368 295 −0.5529 0.4387 0.0877 0.4368 296 −0.5529 −0.4387 −0.0877 0.4368 297 0.0877 −0.5529 0.4387 0.4368 298 0.0877 0.5529 −0.4387 0.4368 299 −0.0877 0.5529 0.4387 0.4368 300 −0.0877 −0.5529 −0.4387 0.4368 301 0.0514 0.5391 0.4611 0.4368 302 0.0514 −0.5391 −0.4611 0.4368 303 −0.0514 −0.5391 0.4611 0.4368 304 −0.0514 0.5391 −0.4611 0.4368 305 0.4611 0.0514 0.5391 0.4368 306 0.4611 −0.0514 −0.5391 0.4368 307 −0.4611 −0.0514 0.5391 0.4368 308 −0.4611 0.0514 −0.5391 0.4368 309 0.5391 0.4611 0.0514 0.4368 310 0.5391 −0.4611 −0.0514 0.4368 311 −0.5391 −0.4611 0.0514 0.4368 312 −0.5391 0.4611 −0.0514 0.4368

The foregoing description and drawings should be considered as illustrative only of the principles of the invention. The invention may be configured in a variety of shapes and sizes and is not intended to be limited by the embodiment shown. In addition, the statements made with respect to one embodiment apply to the other embodiments, unless otherwise specifically noted. For example, the statements regarding FIG. 2(a) with respect to size, shape and geometry apply equally to the embodiments of FIGS. 3, 7-9, 12. It is further understood that the description and scope of invention apply equally (though the descriptions have not been repeated) for each structure that is the same or similar between each of the various embodiment, and whether or not those structures have been assigned a similar reference numeral.

Numerous applications of the invention will readily occur to those skilled in the art. Therefore, it is not desired to limit the invention to the specific examples disclosed or the exact construction and operation shown and described. Rather, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention. 

1. A golf ball comprising: a body having an outer shell with an outersurface; and a pattern formed in the outer surface of said body, the pattern comprising a polyhedron having a plurality of flat faces, each of said plurality of flat faces having one or more sharp edges.
 2. The golf ball of claim 1, wherein said plurality of faces are circumscribed in a sphere, wherein only the sharp corners forming vertices of the polyhedron lie on the sphere.
 3. The golf ball of claim 2, said sphere having a diameter of at least 1·68 in.
 4. The golf ball of claim 1, wherein at least one of the plurality of faces of the polyhedron contains one or more dimples.
 5. The golf ball of claim 1, wherein said plurality of faces are each in a plane.
 6. The golf ball of claim 1, wherein said plurality of faces are contiguous to touch one another and form a single continuous outer surface of said body.
 7. The golf ball of claim 1, wherein said plurality of faces are at an angle with respect to one another to define said one or more sharp edges and said one or more sharp corners.
 8. The golf ball of claim 1, wherein said pattern comprises a Goldberg polyhedron.
 9. The golf ball of claim 1, wherein said plurality of faces comprise a plurality of first faces having a first shape and a plurality of second faces having a second shape.
 10. The golf ball of claim 9, wherein said first shape comprises a pentagon and said second shape comprises a hexagon.
 11. The golf ball of claim 9, wherein said plurality of first faces comprise twelve and said plurality of second faces comprise
 150. 12. The golf ball of claim 9, wherein a ration of said plurality of first faces to said plurality of second faces comprises 12.5:1.
 13. The golf ball of claim 1, wherein said plurality of flat faces having one or more sharp corners.
 14. The golf ball of claim 1, wherein said edges are linear.
 15. The golf ball of claim 1, wherein two neighboring flat faces form an angle substantially less than 180 degrees.
 16. The golf ball of claim 1, wherein said sharp edges have a radius of curvature that is less than 0.001 D, where D is the diameter of a circumscribed sphere of said golf ball.
 17. A method of forming a golf ball, comprising: forming an outer surface; and, forming a pattern in the outer surface, the pattern having a plurality of flat surfaces defining sharp edges and points therebetween. 